2.4 Continuity
Objective:
Given a graph or equation, examine the continuity of a
function, including left-side and right-side continuity.
Then use laws of continuity to evaluate a limit.
A function is continuous if you can draw it in one motion
without picking up your pencil.
Conditions of continuity at a point wher e x  c :
a. f (c) is defined. (a y - value exists at x  c)
b. lim f ( x) exists. ( the limit exists on right and left)
x c
c. lim f ( x)  f (c) (the y - value is the same as the limit)
x c
Determine whether the function is
continuous at each of the following
locations. Explain.
1.
2.
3.
4.
5.
x=0
x=1
x=2
x= 4
Over what interval(s) is f
continuous?
Three types of Discontinuity – removable, jump, non-removable
A. Removable Discontinuity (hole)
(You can fill the hole by redefining the function at one point)
B. Jump Discontinuity (left and right limits are not equal)
Three types of Discontinuity – removable, jump, non-removable
C. Non-removable “Infinite” Discontinuity (asymptote)
5
4
3
2
1
-5 -4 -3 -2 -1 0
-1
1
2
3
4
5
-2
-3
-4
-5
– the limit is infinite as c approaches c on one or both sides
Left-Continuous and Right Continuous – what does it mean?
Section 2.4, Figure 14
Page 69
Example 1 :
Let f ( x)  x . Compute the left - and right - hand limits
at x  0, 1, and 2. Then determine whether f ( x) is left - continuous ,
right - continuous , or continuous at these points.
Example 2 :
Compute the left - and right - hand limits at x  0, 1, and 2.
Then determine whether f ( x) is left - continuous ,
right - continuous , or continuous at these points.
Example 3 :
Find the value of the contant(s) that makes the piecewise function
continuous .
 2 x  9 x 1
f ( x)  
 4 x  c
for x  3
for x  3
Laws of Continuity – page 64
Laws of Continuity – page 64
The Laws of Continuity are intended to help us
determine the continuity of functions containing
sums, products, quotients, multiples, powers,
and roots without looking at a graph.
Knowing that a function is continuous allows us
to use direct substitution to evaluate a limit.
Evaluate each limit using the laws of continuity .
1. lim tan x
x

4
2. lim 4  x 2
x 0
3. lim sin 2 ( sin 2 x)
x

3
 x  1, x  0
4. lim f ( x) if f ( x)   2
x 0
 x  1, x  0