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Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.

I'm not too sure that this question is correct though, as I think I have a counterexample. Let S be the union of (-1,0) and (0, 1), then clearly S is open. Define f(x) = |x| / x for x in S.

Then if x < 0, f(x) = -1, and if x > 0, f(x) = 1.

f'(x) = 0 at every x in S, since 0 is not in S, so f is differentiable on S and the derivative is bounded.

And now f is not uniformly continuous on S, since if we set ε= 1, let δ be arbitrary, and pick x,y close to 0 such that x<0, y>0, and |x - y| < δ, it does not follow that |f(x) - f(y)| < ε. So no δ will work for this ε.

I'd really appreciate any feedback on my reasoning. Thanks for your time!