Continuity on Closed and Half-Closed Intervals Examples

Let 

Determine whether f is continuous on each of the following intervals:

• [10,12]

• [10,12)

• (10,12]

• (10,12)

First, we'll draw the function:

Now it's easier to see what's going on.

• [10,12]: f is not continuous on this interval because f(12) is undefined.

• [10,12): f is continuous on this interval. For starters, f is continuous on (10, 12). While it's true that if we look at the whole graph, f is discontinuous at x = 10, but for the sake of continuity on [10,12), we only care about the right-sided limit. Since , f is continuous on [10,12).

• (10,12]: f is not continuous on this interval because f(12) is undefined.

• (10,12): f is continuous on this interval.