The purpose of this problem is to create a continuous function f : [0.1] rightar

Question

The purpose of this problem is to create a continuous function f : [0.1] rightarrow R such that f(0) = 0 and f(1) = 1, but f'(x) = 0 if z C. the Cantor set, reminder that the Cantor set C = has "length" 0. As usual, It just seems wrong that a continuous function can be im increasing from f(x) = 0 to f(1) = 1 but the derivative f'(x) - 0 almost everywhere. The basic idea is to define a sequence of continuous functions fn whore fn increases from 0 to 1 on [0, 1j and is constant on all submits that have boon removed to form Cn, Specifically, define the sequence (fn) iteratively by Let f0(x) = x and for all c ge 1 define Prove that each fn is continuous and increasing on [0.1], Prove that the sequence of function (fn) converges uniformly on [0,1]. Let f = Iim fx fn. Prove that f is continuous and increasing on [0, 1] and that f(0) = 0 f (1) = 1. Prove that f'(x) = 0 for all z [0, 1]/c.

Explanation / Answer

Hi I have done the solution for this in my notebook. But the solution is too big , I cannot type it here because very less time is remaining.. So please rate me Lifesaver and I'll share the answer with you through email or cramster inbox. I don't do this generally, but I have no other option here because there's very less time left... you need not worry as I have the solution ready

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