Suppose that f:R--> R is differentiable on R and that the derivative of f is bounded on R. Prove that f is uniformly continuous on R.
Three words: MEAN VALUE THEOREM
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Let and you're done.
If , then is said to be Lipschitz continuous, with the Lipschitz constant being the smallest such that satisfies that inequality. Lipschitz continuity always implies uniform continuity, but not necessarily the other way around.