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Counterexamples around differentiation of sequences of functions

We consider here sequences of real functions defined on a closed interval. Following theorem is the main one regarding the differentiation of the limit.

Theorem: Suppose (fn) is a sequence of functions, differentiable on [a,b] and such that (fn(x0)) converges for some point x0[a,b]. If (fn) converges uniformly on [a,b], then (fn) converges uniformly on [a,b] to a function f and for all x[a,b] f(x)=limnfn(x) What happens if we drop some hypothesis of the theorem?

If (fn) converges uniformly to f, and the functions fn are differentiable, does this imply that f is also differentiable?

The answer is no. An example is the sequence of functions
fn:[1,1]Rx1n2+x2 These functions are differentiable according to the chain rule as gn(x):x1n2+x2 is strictly positive on [1,1]. We also have x2f2n(x)=1n2+x2(|x|+1n)2 hence
|x|fn(x)|x|+1n as fn is positive. So by the squeeze test, (fn) converges uniformly to the absolute value function f(x)=|x|. But this function is not differentiable at 0. Thus, the uniform limit of differentiable functions need not be differentiable.

If (fn) converges uniformly to f, and the functions fn are differentiable, does (fn) also converges?

The answer is again negative as we can see considering the functions fn:[0,2π]Rxsinnxn As |siny|1 for yR we get for x[0,2π]
0|fn(x)|1n Hence (fn) converges uniformly to the always vanishing function f(x)=0. On the other hand, fn(x)=ncosnx. And so in particular fn(0)=n does not converge. Consequently, (fn) does not converge pointwise (nor uniformly).

And if (fn) converges, is the limit f?

Again not. Consider fn:[1,1]Rxx1+n2x2 On one hand, we have for all x[1,1], |fn(x)||x| as 1+n2x2>1. Hence |fn(x)|1n for |x|1n.

On the other hand for |x|1n, n2|x|n therefore
|fn(x)|1n2|x|1n Finally for all x[1,1] |fn(x)|1n and (fn) converges uniformly to the function f(x)=0 using the squeeze test.

fn is differentiable with
fn(x)=1nx2(1+nx2)2 For all n1, fn(0)=1 and for x0
|fn(x)|1+nx2(1+nx2)2=11+nx2n0 Conclusion: (fn) is a sequence of differentiable functions that converges uniformly to the zero function whose sequence of derivatives converges to a function different that the derivative of the limit.

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