Continuity of Functions
Topics
Introduction to Continuity Of Functions  At A Glance:
When looking at continuity on an open interval, we only care about the function values within that interval.
If we're looking at the continuity of a function on the open interval (a,b), we don't include a at b, they aren't invited. No value of x less than a or greater than b is invited, either. This is an exclusive club.
In a closed interval, denoted [a,b], we also must also invite our friends a and b to the pool party. Halfclosed intervals either invite a, [a,b), or b, (a,b]. To talk about continuity on closed or halfclosed intervals, we will see what this means from a continuity perspective. Start with a halfclosed interval of the form [a,b). What does it mean for a function to be continuous on this interval?
Since we can only approach a from the right, we use the continuity definition for a rightsided limit instead of a two sided limit. We say f is continuous on [a,b) if f is continuous on (a,b) and
 f(a) exists
 exists
 f(a) and agree
Sample Problem
This function is continuous on the interval [a,b):
This function is continuous on (a,b). f(a) is defined, is defined, and the function value at a agrees with the rightsided limit at a.
Sample Problem
The following functions are not continuous on the interval [a,b).
This function is not continuous on [a,b) because f(a) is undefined. We could also say this function is not continuous on [a,b) because does not exist.
This function is not continuous on the interval [a,b) because .
This function is not continuous on the interval [a,b) because it is not continuous on the open interval (a,b).
Example 1
Let Determine whether f is continuous on each of the following intervals:

Exercise 1
What must be true for a function to be continuous on an interval of the form (a,b]?
Exercise 2
What must be true for a function to be continuous on an interval of the form [a,b]?
Exercise 3
Determine if f is continuous on each of the following intervals.
 [a,b]
 [a,b)
 (a,b]
 (a,b)
Exercise 4
Determine if f is continuous on each of the following intervals.
 [a,b]
 [a,b)
 (a,b]
 (a,b)
Exercise 5
Determine if f is continuous on each of the following intervals.
 [a,b]
 [a,b)
 (a,b]
 (a,b)
Exercise 6
Determine if the function
Exercise 7
Determine if the function
is continuous on each of the following intervals.
 [0,1]
 [0,1)
 (0,1]
 (0,1)
 [1,5]
 [1,5)
 (1,5]
 (1,5)