I am working on a concept of uniform differentiability. f is uniformly differentiable on [a,b] if ∀ ε ∃ a δ such that |(f(x+h)-f(x))/h - f'(x)| < ε for all x in the interval when 0<|h|<δ. I am working on a theorem on it. If f has a continuous derivative on a closed interval, f is uniformly differentiable in that interval. I need a proof that doesn't use the mean value theorem. An elementary proof so to speak. By contradiction and utilizing Weierstrass's limit point theorem
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From this post it seems like uniform differentiability is a known, but rarely used concept. – Nikolaj Pedersen Dec 26 '21 at 20:00
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@NikolajPedersen And this post says it's equivalent to a very commonly used concept. – Arthur Dec 26 '21 at 20:06