Download presentation

Presentation is loading. Please wait.

Published byAna Ireton Modified over 2 years ago

1
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.

2
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!”

3
Examples of Discontinuity: 1. 2. 3.

4
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…

5
Example: Greatest Integer Function: At what points is this graph discontinuous?

6
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.

7
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).

8
Example: Discuss the continuity of. Discuss the continuity of.(Graph.)

9
Example: Find the limit (if it exists). Discuss the continuity of the graph. a. b.

10
Example:

11
Find the x-values (if any) where f(x) is not continuous. Label as removable or non- removable discontinuities. a. b.

12
Example:

13
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:

14
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.

15
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 )

16
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

Similar presentations

OK

Informal Description f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

Informal Description f(x) is continuous at x=c if and only if there are no holes, jumps, skips or gaps in the graph of f(x) at c.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on the nationalist movement in indochina Ppt on forward rate agreement quotes Ppt on schottky diode leakage Ppt on solar power energy Ppt on 60 years of indian parliament news Ppt on hindu religion and cows Ppt on main idea and supporting detail Book appt online Ppt on partnership act 1932 Ppt on life study of mathematician salary