Can any continuous function be re-parameterized into a differentiable function?

There is a broad class of functions for which this reparametrization works at a point, say the origin. A function $f$ is in $C^\alpha$, $0<\alpha$, when $$|f(x)-f(y)|\le k|x-y|^\alpha$$

Let $g_n(x):=x^n$ for $n$ odd, and $g_n(x):=\begin{cases}x^n&x\ge0\\-x^n&x<0\end{cases}$ for $n$ even.

Given $f\in C^\alpha$, let $n>1/\alpha$ , then $$\left|\frac{f(g_n(x))-f(g_n(0))}{x}\right|\le k\frac{|g_n(x)-g_n(0)|^\alpha}{|x|}= k|x|^{n\alpha-1}\to0,\quad x\to0$$ so $f(g_n(x))=f(0)+o(x)$.

This means functions like $|x|^{\beta}$, $\beta\ge0$, and the Weierstrass functions, can be smoothed out (at one point). But functions like $1/\ln|x|$, though continuous, cannot be smoothed with this reparametrization (though it can with $g(x)=sgn(x)e^{-1/|x|}$).