If a function $f: [0,1] \to \mathbb R$ is continuous, does it mean that it's uniformly continuous?
I am not sure how to justify this in a quite formal way. My idea is that it does not necessarily have to be the case - even though the interval is bounded, we can still imagine a function with a very rapidly increasing slope. But, on the other hand, the interval is bounded, therefore we can't have infinitely large slopes and so the slope of every function is bounded (Lipschitz condition) therefore I'd be rather more inclined to say that this is, in fact, the case. What do you think?
It is a theorem that continuous functions over compact sets are uniformly continuous.
Note also that uniformly continuous is not equivalent to have a bounded derivative (although this implies uniform continuity). Even more, a function can be continuous at every point of it's domain and not be differentiable at any of them.