A 4 CONTINUITY Investigating Continuity Continuous Function A

A. 4 CONTINUITY

Investigating Continuity ■ Continuous Function – A function with a connected graph.

Definition of Continuity ■

Continuity ■ function is continuous on an interval IFF it is continuous at every point of the interval. ■ A continuous function is one that is continuous at every point IN ITS DOMAIN. ■ If a function is not continuous at a point, that is known as a point of discontinuity.

Types of Discontinuities NON REMOVABLE ■ Jump Discontinuity – The one-sided limit exists, but have different values

Types of Discontinuities NON REMOVABLE ■ Infinite Discontinuity ■ Let's say you have a function like f(x) = 1/x. – Then, as x goes to 0 from the right (x > 0), the function goes toward positive infinity. – As x goes to zero from the left (x < 0), the function goes toward negative infinity. – At x = 0, the function has no defined value. – We say that x = 0 is an infinite discontinuity, because the limits around the undefined point are infinite.

Types of Discontinuities NON REMOVABLE ■ Oscillating Discontinuity ■ An oscillating discontinuity occurs at a value of x near to which a function refuses to settle down. ■ The function y = sin (1/x) has a discontinuity at x = 0 because it is not defined at x = 0.

Types of Discontinuities ■ Removable discontinuity ■ A point on the graph that is undefined, or does not fit in with the rest of the graph. A “hole” in the graph. ■ The right-handed limit and the left-handed limit must be the same around the point of discontinuity. ■ A discontinuity is removable if f can be made continuous by appropriately defining (or redefining) f(c).

Identify each point of discontinuity and type

Continuity on an open interval ■ A function is continuous on an open interval (a, b) if it is continuous at each point in the interval.

Continuity on a closed interval ■

Continuity of a composite function ■

Intermediate Value Theorem ■ If f is continuous on the closed interval [a, b] and w is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = w.

Determining Continuity Algebraically ■ A function has a discontinuity at any point that makes the function undefined or indeterminate form on its domain.

Example 1 ■ Use the graph to determine the limit, and discuss the continuity of the function. = 1 = 2 = DNE

Example 2 ■ Use the graph to determine the limit, and discuss the continuity of the function. = 2 = 2 f(1) = 3

Example 3 ■ Discuss the continuity of the function The function is continuous at every value of x except x = 2.

Example 4 ■ Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable?

Example 5 ■

Example 6 ■ 2 2

Example 7 ■ 14 11 DNE

A. 4 Practice Problems ■ Pg. 78 -81 #2, 4, 6, 15, 17, 18, 25, 36, 40, 61, 63, 89 ■ Pg. 79 # 15 (decide if the function is continuous at x = 3) ■ Pg. 79 #17 (decide if the function is continuous at x = 1) ■ Pg. 79 # 18 (decide if the function is continuous at x = 1)