Continuity on Closed and Half-Closed Intervals - 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 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, b) 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.