Let W(x) be a real-vlued function defined on a (possibly infinite) interval T⊆R containing 0 that is continuous everywhere differentiable nowhere on T.
Define the sequence of function fn:T→R as follows:
f0(x)≜ and f_n(x)\triangleq\int_{0}^{x}f_{n-1}(u)du for n=1,2,3,4,5,... .
Then it follows the fact that f_n is differentiable exactly n-times everywhere on \text{T}.
Question:
Does there exist such W(x) and \text{T} so that the sequence of function f_n on \text{T} "converges" to a limit function f_{\infty} on \text{T} with f_{\infty} being infinitely many times differentiable (i.e., smooth) on \text{T}?
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