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?

  • $\begingroup$ There are even polynomial functions... $\endgroup$ – Orest Bucicovschi Aug 23 '17 at 15:27

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.


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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