Section 2.5 Notes Page 1
2.5 Continuity
Continuity: graph is connected with no breaks or holes.
Let f(x) be a graph. Continuity at x = c occurs if all three of the following are true:
1.) f (x0)is defined. (There are no vertical asym ptotes or holes at x x0)
2.) lim f (x) exists. (If it didn’t exist, then there must be a break in the graph or a vertical asymptote.)
xx0
3.) lim f (x) f (x ) (I can plug x in for x since there is not a hole).
xx0 0 0
Let’s look at the below graph from the previous section. Can you tell which places the graph is discontinuous?
What value should be assigned to f(2) to make the function continuous at x = 2?
We see that the graph will be discontinuous at x = -2, 0, 2, and 4. .
The a sboe½ld vfalue ssigned to f(2) to make it continuous.
EXAMPLE: Describe the set of x-values where the function is continuous, using interval notation:
1
f (x) (2 x)5
For all these problems you want to look for any place where the equation will be undefined. This only occurs if
you are dividing by a negative number or taking the even root of a negative number. In this problem there are
no places where we are dividing by zero, and we have an odd root here since f(x) can be written as:
f (x) 2 x . Therefore, it will be continuous atevery value of x. So our answer is ,.
EXAMPLE: Describe the set ofx-values where the function is continuous, using interval notation:
f (x) 6x 35
We are only allowed to take the square root of positive numbers or zero. Therefore this is the only place where
the graph is continuous. So to solve this one, we will let everything inside the square root be greater than or
equal to zero: 6x 35 0 . Solving this will give us x 35 . In interval notation the answer is:35 , .
6 6 Section 2.5 Notes Page 2
EXAMPLE: Describe the set of x-values where the function is continuous, using interval notation:
f x 1
x 1
In the problem above, since no x value will make the bottom zero then this graph will be continuous for all x.
So our answer is ,.
EXAMPLE: Describe the set of x-values where the function is continuous, using interval notation:
f (x) 4x 5 cos(2x)
There is no place where there could be a division of zero or square root of a negative number. Therefore, this is
continuous at all values of x. Therefore our interval is: ,.
EXAMPLE: Describe the set of x-values where

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