# Continuity on Closed and Half-Closed Intervals

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* and; they aren't invited. No value of *x* less than *a* or greater than *b* is invited, either. This is an exclusive club, with the parentheses serving as the bouncers.

In a closed interval, denoted [*a*, *b*], we're lowering our standards a bit by inviting *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'll 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*,* <em>b</em>*) 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 as well.

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*).

It's all just like continuity on open intervals. The only difference is that now we have to check the endpoints.