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I need to show that the set of continuous nowhere differentiable functions on $[0,1]$ are dense in $C([0,1])$. I've got the following hints on how to do that: Split $$C([0,1])=\bigcup_{n\geq 0}A_n\,\cup\,\bigcup_{n\geq 0}B_n\,\mathring{\cup}\,C$$ with $$A_n\,=\,\{f\in C([0,1])\,|\,\exists x\in [0,1-\frac{1}{n}]\,:\,\frac{|f(x+h)-f(x)|}{h}\leq n,\,\forall h\in (0,\frac{1}{n}]\}$$ and $$D_r\,=\,\{f\in C([0,1])\,|\,f \text{ is differentiable from the right side for at least one } a\in[0,1)\}$$.

First I should show, that $A_n$ is closed $\forall n\in\mathbb{N}$. I've allready done this. Next I should show, that $\mathring{A_n}=\emptyset\,\forall n\in\mathbb{N}$. Someone told me that I should use the Weierstraß-Approximation-Theorem and come to a contradiction, but I have no clue how to do that.

Lastly I should define $B_n$ and $D_l$ analogously for the left side derivative and conclude that $\bigcup_{n\geq 0}A_n\,\cup\,\bigcup_{n\geq 0}B_n$ is meager. Any help would be greatly appreciated.

Fluadl
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  • I've already considered it, but in my problem I should show that $\mathring{A_n}$ is empty. I don't see where this is addressed in the other post. Is this indirectly shown? – Fluadl Apr 28 '24 at 15:44
  • The continuously differentiable functions $C^1[0,1]$ are dense in $C[0,1].$ Let $h$ be a continuous nowhere differentiable function on $[0,1].$ The set $${f+ah,:, f\in C^1[0,1],\ a\neq 0}$$ is dense in $C[0,1]$ and consists of nowhere differentiable functions. So it suffices to find one nowhere differentiable function $h.$ – Ryszard Szwarc Apr 28 '24 at 18:58

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