Can we approximate a smooth function by a continuous and nowhere smooth function uniformly

Question

From Stone-Weierstrass approximation theorem we know that we can approximate a continuous(no matter differentiable or not) by a polynomial function uniformly within a compact interval domain.
But,can we approximate a smooth function by a nowhere smooth continuous function uniformly?

Answer 1

Yes. Let $g:[-1,1]\to\mathbb{R}$ be a given smooth function. Let $g_n:[-1,1]\to\mathbb{R}$ be a sequence of polynomial that converges uniformly to $g$ and let $h:[-1,1]\to\mathbb{R}$ be a bounded nowhere differentiable function. Then the sequence $$v_n (t)=g_n (t) +n^{-1} h(t) $$ is a sequence of nowhere differentiable functions that converges to $g$ uniformly.