ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
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Activity C2 – Continuity of a Function
In activity C1 we looked at the graph of a function to find its points of discontinuity.
Determining where a function is discontinuous given its equation (no graph) is a little
more difficult.
If we know how to find the domain of basic functions, we can use the theorem(RS2) at
right to determine where these basic functions are continuous.
Find the intervals of continuity for the following functions.
1. f ( x)  3x 2  x  1
2. f ( x) 
x2 1
x2
3. f ( x)  2 x  5
Since we know how to find intervals of continuity for individual functions we can now
find the points of discontinuity of a piecewise defined function. This process is more
complicated because we need to investigate the continuity of the functions within the
piecewise function, and also the continuity of the functions at the endpoints of the
domain’s intervals. The examples(RS3) at right shows how to find the points of
discontinuity of a piecewise defined function.
4. Answer the questions that follow given
2 x  1 x  2
f ( x)  
x2
 x
a) Find the points of discontinuity, if any, for y  2 x  1 .
b) Find the points of discontinuity, if any, for y   x .
c) Find lim f ( x ) .
x2
d) Find f (2) .
e) Is f ( x) continuous at x = 2.
f) Find the interval(s) of continuity for f (x).
Some problems can be more difficult as shown in the example(RS4) at right.
5. Answer the questions that follow given
ACOW LIMITS AND CONTINUITY MODULE
Updated 5/28/2016
Page 2 of 2
 x 2 x  1

f ( x)   4
x  1

x3
a) Find the points of discontinuity, if any, for y  x 2 .
b) Find the points of discontinuity, if any, for y 
4
.
x3
c) Find lim f ( x) .
x 1
d) Find f ( 1) .
e) Is f ( x) continuous at x = –1?
f) Find the interval(s) of continuity for f (x).
6. Find all points of discontinuity for the following functions.
 x x  4

a) f ( x)   4  4  x  2
3 x 2 x  2

 1
 x  5 x  2
b) f ( x)  
1
x  2
 x