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Take the metric space $(C([0,1]),d)$ with $C([0,1])$ the set that contains all continuous functions $f: [0,1] \to \mathbb{R} $ and the metric $d(f,g) = \sup \{|f(x)-g(x)| : x \in [0,1]\}$.

Now, I was wondering that if we take a non differentiable, continuous function $f$ in this metric space, are there any differentiable functions in the set $B(f,\varepsilon) = \{ g \in C([0,1]) : d(f,g) < \varepsilon \}$?

I do think that this is true but that is just my intuition, can someone help me find a solution?

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  • $\begingroup$ There are even polynomial functions... $\endgroup$ – Orest Bucicovschi Aug 23 '17 at 15:27
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Your question is equivalent to asking

  • Is it possible to approximate a continuous function uniformly with a differentiable one, or
  • Is the set of differentiable function dense in $C([0,1])$.

Both is true and stays true if you replace "differentiable" by differentiable infintely often and can be seen by convolution with mollifiers.

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That is true. In fact, one may even take the function $g$ to be a polynomial. See the Stone-Weierstrass approximation theorem.

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