Let f:R->R, suppose there is such that for all
show that f is continuos.
Have you made any attempt? Have you seen this condition before? It's known as the Lipschitz condition. It comes up a lot in analysis and topology.
It is powerful as it specifies a condition on a function which is stronger than being continuous but not as strong as being differentiable (comes in handy for Picard's Theorem and the study of weak solutions of PDEs)
I recommend googling it and then attempting the question yourself.
If you get stuck here is a pointer in the right direction:
Spoiler:
Hope this helps.
pomp.