5 Steps to a 5 AP Calculus BC 2019

64 STEP 4. Review the Knowledge You Need to Score High

5.3 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. xlim→af(x) exists


3. xlim→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 functions f andg are continuous ata, then the functions f+g, f−g, f ·g
   and f/g,g(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 function f 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 5.3-1.)

[–5,5] by [–10,10]

Figure 5.3-1