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 $\text{T}\subseteq\mathbb{R}$ containing $0$ that is continuous everywhere differentiable nowhere on $\text{T}$.

Define the sequence of function $f_n:\text{T}\to\mathbb{R}$ as follows:

$$f_0(x)\triangleq{W(x)}$$ 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 $\text{T}$?