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# Continuity of Functions

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# 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* 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*).