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I'm trying to prove a theorem in this page : http://en.m.wikipedia.org/wiki/Differentiation_under_the_integral_sign#/search

Everything is completely fine except one line.

How do i prove that "For all $\epsilon>0, (\exists \Delta \alpha)\forall x\in[a,b], |f(x,\alpha+\Delta \alpha)-f(x,\alpha)|<\epsilon$?

It's written there that it's a consequence of Heine-Cantor theorem. But how?

Let's fix $\alpha$ and $\epsilon$.

Then, there exists $\delta>0$ such that $d(x,y)<\delta \Rightarrow |f(x,\alpha)-f(y,\alpha)|<\epsilon$

This is strictly weaker than the statement written above. How could i prove that??

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$f$ is continuous ant thus on the bounded rectangle under consideration uniformly continuous as a function in both variables, so the stated inequality is also true for variations in the second variable.

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