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
- f(a) and agree
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.
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.