In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}.(adsbygoogle = window.adsbygoogle || []).push({});

What I don't like about the theorem is the huge amount of assumptions required:

" Suppose that {fn} is a sequence of functions which are differentiable on [a,b], with integrable derivatives fn', and that {fn} converges (pointwise) to f. Suppose, moreover, that {fn'} converges uniformly on [a,b] to some continuous function g. Then f is differentiable and f'(x) = lim n-->infinity fn'(x). "

Are really EACH of these assumptions necessary for this to be true? Are there counterexamples for any combination of missing hypotheses? With all these assumptions the proof is quite easy, and I suspect this might be the reason, but in this case, how many of these assumptions can we get rid of?

I've seen the counterexample of fn = sqrt(x^2+1/n^2), which converges uniformly to |x|, which is not differentiable. And also fn = 1/n*sin(n^2 x) which converges to 0 but the derivatives of fn do not always converge.

But what about counterexamples involving non-integrable derivatives, non-uniform convergence to a continuous g, or uniform convergence to a function g which is not continuous? And doesn't uniform convergence of the derivatives imply at least pointwise convergence of the functions? etc, etc... I think you get my point (no pun intended)...

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# Uniform convergence and derivatives question

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