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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?

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    $\begingroup$ The question in the title is different from the question in the question body $\endgroup$ – Alessandro Codenotti Aug 18 at 14:19
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    $\begingroup$ thanks for noticing,I have changed it. $\endgroup$ – user685348 Aug 18 at 14:33
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    $\begingroup$ Can you construct a continuous bounded nowhere smooth function ? $\endgroup$ – reuns Aug 18 at 14:41
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    $\begingroup$ Nowhere differentiable functions are dense in the space of continuous functions. $\endgroup$ – Matt Samuel Aug 18 at 17:59
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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.

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