Uniform Continuity ? How can you tell?

Question

Let $f(x) = \dfrac{1}{1+e^x}$.

Is f uniformly continuous?

In general, how can you recognize if a function is uniformly continuous?

Answer 1

Here are two conditions which will help you in most cases:

• If a function is Lipschitz-continuous, it is in particular uniformly continuous. As noted in the comments, this follows e.g. if $f$ is differentiable and $|f'|$ is bounded (and works in your case).
• If a function defined on a compact set is continuous, it is automatically uniformly continuous. This also implies that continuous periodic functions are uniformly continuous.

Edit: Your derivative is certainly bounded ;)