Is it true that for all functions h, if lim𝑥→𝑎+ℎ(𝑥) exists and lim𝑥→𝑎−ℎ(𝑥) exists, then h is continuous at a?

Answer 1

Hi Naeema,

In short, no. That statement is false.

Consider a function with a removable discontinuity at x=a (a hole in the function at one point where x=a).

The function h(x) can have a limit that exists as "x" approaches "a" from both positive and negative directions, but the point "a" could be a removable discontinuity, and therefore not continuous.

So you can't say that it is necessarily continuous only because it has limits from both sides, that is false.

Hope this helps! :)