Wednesday, 14 June 2017

real analysis - Turning a continuous everywhere differentiable nowhere function into a smooth function by infinitely many times definite integration?

Let W(x) be a real-vlued function defined on a (possibly infinite) interval TR containing 0 that is continuous everywhere differentiable nowhere on T.



Define the sequence of function fn:TR 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}?

No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...