I am reading the paper Coordinate Descent Algorithms by Wright (see here) and in Assumption 1 on page 12 it is assumed that the objective function $f$ is uniformly Lipschitz continuously differentiable. I find this assumption confusing. It seems that Lipschitz continuity implies uniform continuity (see here) so the word uniformly is superfluous here, right? Are there any examples of Lipschitz continuous functions that are not uniformly continuous? Is this just a mistake or is there some meaning of this assumption?
Any help is much appreciated!
Every continuously differentiable function is locally lipschitz. However, the function $f(x)=e^x$ is continuously differentiable, but not uniformly lipschitz.
So we are essentially assuming that the derivative exists and is globally bounded.