5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 7, 2018 9:52

Limits and Continuity 99

6.3 Continuity of a Function

Main Concepts:Continuity of a Function at a Number, Continuity of a Function over an

Interval, Theorems on Continuity

Continuity of a Function at a Number

A function f is said to be continuous at a numberaif the following three conditions are

satisfied:

1. f(a) exists


2. limx→af(x) exists


3. limx→af(x)= f(a)

The functionf is said to be discontinuous ataif one or more of these three conditions are

not satisfied andais called the point of discontinuity.

Continuity of a Function over an Interval

A function is continuous over an interval if it is continuous at every point in the interval.

Theorems on Continuity

1. If the functionsfandg are continuous ata, then the functionsf +g, f −g, f ·g,
   f/g, andg(a)=0, are also continuous ata.


2. A polynomial function is continuous everywhere.


3. A rational function is continuous everywhere, except at points where the denominator
   is zero.


4. Intermediate Value Theorem: If a functionf is continuous on a closed interval [a,b]
   andkis a number withf(a)≤k≤f(b), then there exists a numbercin [a,b] such
   thatf(c)=k.

Example 1

Find the points of discontinuity of the functionf(x)=

x+ 5

x^2 −x− 2

.

Since f(x) is a rational function, it is continuous everywhere, except at points where

the denominator is 0. Factor the denominator and set it equal to 0: (x−2)(x+1)=0.

Thusx=2orx=−1. The function f(x) is undefined atx=−1 and atx=2. There-

fore, f(x) is discontinuous at these points. Verify your result with a calculator. (See

Figure 6.3-1.)

[−5, 5] by [−10, 10]

Figure 6.3-1