$G_\delta$ set of nowhere differentiable functions? I'm trying to show that in $C([0,1])$ with the supremum metric, there exists a dense $G_\delta$ set of nowhere differentiable functions. Honestly, I don't know how to approach this. Any help would be greatly appreciated. 

The density of nowhere-differentiable functions is a rather deep result, and a classical application of the Baire category theorem. Here is a nice treatment (see proposition 3). To express the nowhere-differentiable functions as a $G_\delta$ set should require only a slight modification of the linked proof.
