Lipschitz vs uniform continuity.

[gottfried]

What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both).

So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and I feel that uniformly continuous functions have the same property since one delta must work for the entire function. A change in the domain of size less than delta must always correspond to a change in the range less than epsilon and this fact makes me feel as though the gradient must also be bounded.

Also what does it mean intuitively about a function if it is holder continuous? Again the definition seems to be saying simply that the gradient is bounded.

So either I'm completely wrong and none of these forms of continuity require a bounded gradient or I'm right but I'm missing some other important information. Help appreciated.

 

[economicsnerd]

It's instructive to think of a function $f:[0,1]\to \mathbb R$ which is uniformly continuous but not Lipschitz continuous, e.g. the square root function. It's continuous, thus (since $[0,1]$ is compact) uniformly continuous. However, any attempt at Lipschitz continuity will fail at $0$.