CONTINUITY
Roughly speaking, a function is said to be continuous on an interval if its graph has no breaks, jumps,
or holes in that interval. So, a continuous function has a graph that can be drawn without lifting the
pencil from the paper.
1. Which of the functions graphed below are continuous?
20
20
15
15
10
10
5
5
0
0
-1
-0.5
-5 0
0.5
1
-4
-3
-2
-1
-5 0
1
2
3
-10
-10
-15
-15
-20
x
-20
x
10
8
8
6
6
4
4
2
2
-5
-3
0
-3
-2
-1
0
x
1
2
0
-1 -2
1
3
5
-4
3
More specifically, a function is continuous if nearby values of the independent variable give nearby
values of the function.
Example: Let p ( x ) be the cost of mailing a letter weighing x ounces. According to our current postal
rates, p(1.00)  $0.37 and p (0.99)  $0.37 . But p (1.01)  $0.59 . So values nearby x  1.00 do not
give nearby values to p(1.00) . This function is not continuous at x  1.00 .
2. Which functions that we have studied so far may not be continuous?
3. Is this function continuous on the given intervals? Why or why not?
y
4
x 3
a. on [0,4]
b. on [-2.5,2.5]
4. Is this function continuous on the given intervals? Why or why not?
g ( x) 
ecos x
sin x
  3 
a. on  , 
4 4 
   
b. on  , 
 4 4