- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
- Determining Continuity
- The Informal Version
- The Formal Version
- Properties of Continuous Functions
- Boundedness
- Extreme Value Theorem
- Intermediate Value Theorem
- In the Real World
- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
- Appendix: Intervals and Interval Notation
Introduction to :
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. Half-closed intervals either invite a, [a,b), or b, (a,b]. To talk about continuity on closed or half-closed intervals, we will see what this means from a continuity perspective. Start with a half-closed 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 right-sided 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 right-sided 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).
Practice:
Let Determine whether f is continuous on each of the following intervals:
|
, f is continuous on [10,12).What must be true for a function to be continuous on an interval of the form (a,b]?
must exist.
.What must be true for a function to be continuous on an interval of the form [a,b]?
and 
must exist
and f(b) must equal 
Determine if f is continuous on each of the following intervals.
- [a,b]
- [a,b)
- (a,b]
- (a,b)
Determine if f is continuous on each of the following intervals.
- [a,b]
- [a,b)
- (a,b]
- (a,b)
and f is continuous on (a,b))Determine if f is continuous on each of the following intervals.
- [a,b]
- [a,b)
- (a,b]
- (a,b)
Determine if the function 
and 
, f will be discontinuous on any interval containing 0, and continuous on any other interval.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)
). This means if x = 1 is the left endpoint of an interval, f can be continuous on that interval.
.
.
