5/28/2018 Section 1.7 Continuity

1/3

Page 1of

ACC, F12

Section 1.7 Continuity

A. Understanding Continuity Intuitively, we think of something as being continuous if it keeps going without any breaks or

interruptions

Similarly, we can think of a continuousfunction as one whose graph we can trace from end-to-endwithout picking up our pencil. This can only happen if the graph does any have any breaks or

interruptions The following are examples continuous functions:

1. () = + 1 2. () = 32 6 3. () = 0.5| + 4|

Notice that these functions contain no holes, jumps, or breaksB. Three-part Definition

More formally, we say that function()is continuous at a point if the following are true:1. ()is defined2. limit as exists3. lim() =()

If any of these conditions fail, we say that()is discontinuousat the point . Examples:

1. Use the 3-part definition of continuity to explain why the function isdiscontinuous at = 1

2. Identify the intervals on which()is continuous

5/28/2018 Section 1.7 Continuity

2/3

Page 2of

ACC, F12

C. Type of Discontinuities There are 4 types of discontinuities. They are called jump, removable, infinite, and oscillating. See the various types of discontinuities in the graphs below:

D. Theorems about continuous functions Polynomials, trig functions, exponential functions, logarithmic functions, root function, and inverse trig

functions are all continuous at every point in their domain.

Ifand are two continuous functions, then , ,, and/are also continuous. Examples:

1. Let() =5 + 40. Determine the point(s) at which()is continuous.

2. The table below shows the rates a gas company charges each month for natural gas. Use the tableto answer the following:

a. Write a piecewise function to represent the monthly charge ()for a customer who uses therms during the summer

() = ____________________________ if 0 __________________________________ if > ___________

b. Sketch the graph of (). Is ()continuous at = 50?

Summer (May September)

Base charge $10

First 50 therms $0.60 per them

Over 50 therms $0.45 per them

5/28/2018 Section 1.7 Continuity

3/3