# Non differentiable, continuous functions in metric spaces.

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?

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

• Is the set of differentiable function dense in $C([0,1])$.
That is true. In fact, one may even take the function $g$ to be a polynomial. See the Stone-Weierstrass approximation theorem.