1.4 - Continuity Objectives: You should be able to… 1. Determine if a function is continuous. 2. Recognize types of discontinuities and ascertain how to remove discontinuities if possible. 3. Apply the Intermediate Value Theorem to continuous functions.

TEST for Continuity/Definition f (x) is continuous at the point x = c if and only if ALL 3 of the following hold: 1.f(c) is defined.(closed circle) 2. exists. 3. (closed circle at limit) **If any of these three fail, then the continuity at x = c is “destroyed!”

Examples of Discontinuity:

Continuity at Endpoints of a Graph on a Closed Interval f(x) is continuous at its left endpoint a if… f(x) is continuous at its right endpoint b if…

Example: Greatest Integer Function: At what points is this graph discontinuous?

Removable Discontinuities a discontinuity at x = c that can be eliminated/removed by appropriately defining (or redefining) f (c). (Open circle where limit exists) Ex.

Non-Removable Discontinuities a discontinuity that can not be removed at x = c even if you attempt to redefine the value of f (c). Classifications of Non-Removable Discontinuities— Jump, Infinite, and Oscillating (and vertical asymptotes).

Example: Discuss the continuity of. Discuss the continuity of.(Graph.)

Example: Find the limit (if it exists). Discuss the continuity of the graph. a. b.

Example:

Find the x-values (if any) where f(x) is not continuous. Label as removable or non- removable discontinuities. a. b.

Example:

Intermediate Value Theorem (IVT) Consider f (x) is a continuous function on [a, b]. If y 0 is between f (a) and f (b), then y 0 = f (c) for some. Graphically:

Intermediate Value Theorem (IVT) Real-life example: A common sense example: a person’s height. Suppose a girl is 5 ft tall on her 13 th birthday and 5 ft 7 in. tall on her 14 th birthday. For any height h between 5 ft and 5 ft 7 in., there has to be a time when her height was exactly h. For ex., there has to be a time between her 13 th and 14 th birthday that she was 5 ft 4 in. This is reasonable because a person’s height is continuous and does not abruptly change from one value to another.

Example: No Calc Use the IVT to show that the polynomial function has a zero on the interval [0, 1]. That is, show that for some value of c, f (c) = 0. For the interval [x 1, x 2 ], f(c) has to be between f(x 1 ) and f(x 2 ). f(x 1 ) < f(c) < f(x 2 ) or f(x 2 ) < f(c) < f(x 1 )

Example: Verify that the IVT applies to the indicated interval and find the value of c guaranteed by the theorem., [0, 3], f(c) = 4