Limits and Continuity for Vector Valued Functions
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Limits Continuity Limits and Continuity for Vector Valued Functions Bernd Schröder Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction 1. Although we mostly use derivatives and integrals in calculus Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction 1. Although we mostly use derivatives and integrals in calculus, all concepts in calculus can formally be traced back to limits. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction 1. Although we mostly use derivatives and integrals in calculus, all concepts in calculus can formally be traced back to limits. 2. So we must start the calculus of vector-valued functions with limits. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction 1. Although we mostly use derivatives and integrals in calculus, all concepts in calculus can formally be traced back to limits. 2. So we must start the calculus of vector-valued functions with limits. 3. However, limits are cumbersome. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Introduction 1. Although we mostly use derivatives and integrals in calculus, all concepts in calculus can formally be traced back to limits. 2. So we must start the calculus of vector-valued functions with limits. 3. However, limits are cumbersome. 4. Thankfully, a lot of calculus of vector valued functions can be reduced to doing calculus with the components. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. The function f is said to converge to L at p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. The function f is said to converge to L at p if and only if, for all x sufficiently close to but not equal to p, Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. The function f is said to converge to L at p if and only if, for all x sufficiently close to but not equal to p, we have that f (x) is close to L. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. The function f is said to converge to L at p if and only if, for all x sufficiently close to but not equal to p, we have that f (x) is close to L. L will be called the limit of the function at p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity All Limits are Defined the Same Way When the input gets close to the place where we take the limit, the output gets close to the limit itself. Recall: Definition. (Informal definition of the limit at a point.) Let p be a real number. The function f is said to converge to L at p if and only if, for all x sufficiently close to but not equal to p, we have that f (x) is close to L. L will be called the limit of the function at p, denoted L = lim f (x). x→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 u lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 u L lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 u L lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write 7 u L lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write ε 7 u L lim~c(t) = ~L. t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. ε 7 u ] JJ L J J J J J J J J t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. ε 7 u ] JJ L J J J J J J J J t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. ε 7 u ] JJ L J J J J ~c(p − δ ) J J J J t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. (Informal definition of the limit for vector valued functions.) The vector ~L is the limit of the vector valued function ~c(t) at p if and only if, for all t that are sufficiently close to but not equal to p, we have that~c(t) is close to ~L. In symbols, we write lim~c(t) = ~L. ε 7 u ] JJ L J J J J ~c(p − δ ) J ~c(p + δ ) J J J t→p Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = . . . . t→p t→p . . cn (t) limt→p cn (t) Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Explanation. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Explanation. ~c(t) −~L Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Explanation. ~c(t) −~L c1 (t) L1 c2 (t) L2 = .. − .. . . cn (t) Ln Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Explanation. ~c(t) −~L c1 (t) L1 c2 (t) L2 = .. − .. . . cn (t) Ln q 2 2 2 = c1 (t) − L1 + c2 (t) − L2 + · · · + cn (t) − Ln Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Proposition. The limit of a vector valued function~c(t) can be taken componentwise. That is, c1 (t) limt→p c1 (t) c2 (t) limt→p c2 (t) lim~c(t) = lim = , . . . . t→p t→p . . cn (t) limt→p cn (t) where the equation is to be read as “the limit on the left side exists if and only if all limits on the right side exist.” Explanation. ~c(t) −~L c1 (t) L1 c2 (t) L2 = .. − .. . . cn (t) Ln q 2 2 2 = c1 (t) − L1 + c2 (t) − L2 + · · · + cn (t) − Ln Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity 3 − 2t2 Example. Compute the limit lim t2 −t t cos(t)−1 t t→0 or show that it does not exist. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity 3 − 2t2 Example. Compute the limit lim t2 −t t cos(t)−1 t t→0 or show that it does not exist. lim t→0 3 − 2t2 t2 −t t cos(t)−1 t Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity 3 − 2t2 Example. Compute the limit lim t2 −t t cos(t)−1 t t→0 or show that it does not exist. lim t→0 3 − 2t2 t2 −t t cos(t)−1 t Bernd Schröder Limits and Continuity for Vector Valued Functions 3 − 2t2 t−1 = lim t→0 − sin(t) Louisiana Tech University, College of Engineering and Science Limits Continuity 3 − 2t2 Example. Compute the limit lim t2 −t t cos(t)−1 t t→0 or show that it does not exist. lim t→0 3 − 2t2 t2 −t t cos(t)−1 t Bernd Schröder Limits and Continuity for Vector Valued Functions 3 − 2t2 t−1 = lim t→0 − sin(t) 3 = lim −1 t→0 0 Louisiana Tech University, College of Engineering and Science Limits Continuity 3 − 2t2 Example. Compute the limit lim t2 −t t cos(t)−1 t t→0 or show that it does not exist. lim t→0 3 − 2t2 t2 −t t cos(t)−1 t Bernd Schröder Limits and Continuity for Vector Valued Functions 3 − 2t2 t−1 = lim t→0 − sin(t) 3 = lim −1 t→0 0 Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a if and only if the following three conditions are satisfied. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a if and only if the following three conditions are satisfied. 1. f (a) is defined. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a if and only if the following three conditions are satisfied. 1. f (a) is defined. 2. lim f (x) exists. x→a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a if and only if the following three conditions are satisfied. 1. f (a) is defined. 2. lim f (x) exists. x→a 3. lim f (x) = f (a). x→a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Continuity is Always Defined the Same Way The limit at the point must be equal to the value at the point. Recall: Definition. Let f be a function. Then f is called continuous at a if and only if the following three conditions are satisfied. 1. f (a) is defined. 2. lim f (x) exists. x→a 3. lim f (x) = f (a). x→a If f is continuous at every point a in the set A, then f is called continuous on A. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. The vector valued function~c(t) is continuous at a Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. The vector valued function~c(t) is continuous at a if and only if all its components are continuous at a. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. The vector valued function~c(t) is continuous at a if and only if all its components are continuous at a. Moreover, a vector valued function is continuous on the set S Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. The vector valued function~c(t) is continuous at a if and only if all its components are continuous at a. Moreover, a vector valued function is continuous on the set S if and only if all its components are continuous on the set S. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Definition. The vector valued function~c(t) is called continuous at a if and only if the following three conditions are satisfied. 1. ~c(a) is defined, 2. lim~c(t) exists, t→a 3. lim~c(t) =~c(a). t→a The vector valued function~c(t) will be called continuous on the set S if and only if~c(t) is continuous at every point in the set S. Proposition. The vector valued function~c(t) is continuous at a if and only if all its components are continuous at a. Moreover, a vector valued function is continuous on the set S if and only if all its components are continuous on the set S. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). I 1 is continuous on (−∞, 2) ∪ (2, ∞) 2−t Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). I 1 is continuous on (−∞, 2) ∪ (2, ∞) 2−t √ I t + 1 is continuous on [−1, ∞). Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). I 1 is continuous on (−∞, 2) ∪ (2, ∞) 2−t √ I t + 1 is continuous on [−1, ∞). The function~c(t) is continuous on the intersection of these domains. Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). I 1 is continuous on (−∞, 2) ∪ (2, ∞) 2−t √ I t + 1 is continuous on [−1, ∞). The function~c(t) is continuous on the intersection of these domains. So it is continuous on [−1, 2) ∪ (2, 3). Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science Limits Continuity Example. Determine where the function~c(t) = ln 9 − t2 1 √ 2−t t+1 is continuous. Every component is continuous on its domain. I ln 9 − t2 is continuous on (−3, 3). I 1 is continuous on (−∞, 2) ∪ (2, ∞) 2−t √ I t + 1 is continuous on [−1, ∞). The function~c(t) is continuous on the intersection of these domains. So it is continuous on [−1, 2) ∪ (2, 3). Bernd Schröder Limits and Continuity for Vector Valued Functions Louisiana Tech University, College of Engineering and Science
