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BASIC IDEA Intuitively, a function is continuous at x = c if you can draw it without lifting your pen from the paper. In the diagram below, the function on the left is continuous throughout, but the function on the right is not. It is "discontinuous" at x = c. CONTINUITY SOURCE: www.portledge.org/homework/jsillat/documents/1.4lecture_000.ppt 2 FORMAL DEFINITION EXAMPLE OF DISCONTINUITY Formally, a function is continuous at x = c on the open interval (a, b) if the following three conditions are met: f(c) is defined lim f ( x) exists x→c lim f ( x ) = f (c ) x →c 3 • F is continuous at x=c if there are no holes, jumps or gaps at c 4 Removable and Nonremovable discontinuity Graphically: Removable discontinuity Nonremovable discontinuity Removable discontinuity f (x) is continuous at c. lim f ( x) = f (c ) x →c 5 Discuss the continuity of the following graph. 6 Continuity at a Point • A function can be discontinuous at a point – The function jumps to a different value at a point – The function goes to infinity at one or both sides of the point, known as a pole 7 8 Discuss the continuity of each of the following functions: f ( x) = The domain of f is all nonzero real numbers. You can conclude that f is continuous at every x-value in its domain. In other words, there is no way to define f(0) so as to make the function continuous at x = 0. 1 x f ( x) = 1 x Nonremovable Discontinuity 9 Discuss the continuity of each of the following functions: 10 • The domain of f is all real numbers except x = 1. Therefore, you can conclude that f is continuous at every x-value in its domain. • At x = 1, the function has a removable discontinuity. • If f(1) is defined as 2, the “newly defined” function is continuous for all real numbers. x2 − 1 f ( x) = x −1 x2 − 1 f ( x) = x −1 11 Removable Discontinuity 12 Discuss the continuity of the function: • The domain of f(x) is all real numbers. • Therefore, you can conclude that the function is continuous on its entire domain. • When a function is continuous from (-∞,∞), f ( x ) = sin x • We say that the function is everywhere continuous. f ( x) = sin x 13 Everywhere Continuous 14 Theorem Properties of Continuity Continuity Theorem • A function will be continuous at any number x = c for which f(c) is defined, when … f(x) is a polynomial f(x) is a power function f(x) is a rational function f(x) is a trigonometric function f(x) is an inverse trigonometric function 15 16 Continuity of a Composite Function Removing a discontinuity: f ( x) = lim x →1 x3 − 1 x2 −1 has a discontinuity at x = 1 . Write an extended function that is continuous at x = 1 . ( x − 1) ( x 2 + x + 1) 1 + 1 + 1 x3 − 1 = lim = x →1 ( x + 1)( x − 1) 2 x2 −1 x3 − 1 x 2 − 1 , x ≠ 1 f ( x) = 3 , x =1 2 = 3 2 Note: There is another discontinuity at x = −1 that can not be removed. 17 18 Properties of Continuous Functions Continuity on an Interval • If f and g are functions, continuous at x = c Then … s ⋅ f ( x ) is continuous (where s is a constant) • The function f is said to be continuous on an open interval (a, b) if f(x) + g(x) is continuous f ( x) ⋅ g ( x) is continuous f ( x) is continuous g ( x) f(g(x)) is continuous – It is continuous at each number/point of the interval • It is said to be continuous on a closed interval [a, b] if 19 – It is continuous at each number/point of the interval and – It is continuous from the right at a and continuous from the left at b 20 Intermediate Value Theorem Continuity on an Interval If a function is continuous between a and b, then it takes • On what intervals are the following functions continuous? on every value between f ( a ) and f ( b ) . g ( x) = sin x f (b) f ( x) = Because the function is continuous, it must take on every y value between f ( a ) and f ( b ) . f (a) x2 −1 x2 − 4 b a 21 Intermediate Value Theorem The Intermiate Value Theorem implies that if f is continuous on [a,b] and k is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = k. 22 → Locating Roots with Intermediate Value Theorem • Given f (a) and f (b) have opposite sign – One negative, the other positive • Then there must be a root between a and b a b 24