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Continuity and One-Sided Limits Lesson 2.4 2 3 4 Intuitive Look at Continuity A function without breaks or jumps The graph can be drawn without lifting the pencil 5 Continuity at a Point A function can be discontinuous at a point • A hole in the function and the function not defined at that point • A hole in the function, but the function is defined at that point 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 Definition of Continuity at a Point A function is continuous at a point x = c if the following three conditions are met 1. f(c) is defined f ( x) exists 2. lim x c f ( x ) f (c ) 3. lim x c x=c For this link, determine which of the conditions is violated in the examples of discontinuity 8 "Removing" the Discontinuity A discontinuity at c is called removable if … If the function can be made continuous by • defining the function at x = c • or … redefining the function at x = c Go back to this link, determine which (if any) of the discontinuities can be removed 9 Which of These is Dis/Continuous? When x = 1 … why or not x 2x 3 f ( x) x 1 2 x2 2 x 3 g ( x) x 1 if x ≠ 1 and g(x) = 6 if x = 1 Are any removable? x2 2 x 3 h( x ) x 1 if x ≠ 1 and h(x) = 4 if x = 1 x3 F ( x) x 1 if x ≠ 1 and F(x) = 4 if x = 1 10 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 11 Properties of Continuous Functions If f and g are functions, continuous at x = c Then … s f ( x ) is continuous (where s is a constant) f(x) + g(x) is continuous f ( x) g ( x) is continuous f ( x) is continuous g ( x) f(g(x)) is continuous 12 One Sided Continuity A function is continuous from the right at a point x = a if and only if lim f ( x) f (a) x a A function is continuous from the left at a point x = b if and only if lim f ( x) f (b) x b a b 13 Continuity on an Interval The function f is said to be continuous on an open interval (a, b) if • It is continuous at each number/point of the interval It is said to be continuous on a closed interval [a, b] if • 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 14 Continuity on an Interval On what intervals are the following functions continuous? g ( x) sin x x2 1 f ( x) 2 x 4 15 Intermediate Value Theorem Given function f(x) • Continuous on closed interval [a, b] • And L is a number strictly between f(a) and f(b) Then … there exists at least one number c on the open interval (a, b) such that f(c) = L f(b) L f(a) a c b 16 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 Try exercises 88, 90, and 92 : pg 100 17 Assignment Lesson 2.4A Page 98 Exercises 1 – 49 odd Lesson 2.4B Page 99 Exercises 51 – 75, 85 – 105 odd 18