In a homework assigment, we were given a certain recursive definition of a space-filling curve $f : [0,1] \mapsto [0,1]^2$ and asked to determine where it is differentiable. My intuition tells me that (as a fractal construction is involved) the curve should be nowhere differentiable.
I think I finally came up with an elementary proof for this particular curve, however it's quite messy with quite some case distinctions necessary.
So looking for a more elegant proof, my question is:
Does the mere fact that $f$ is a space-filling curve, i.e. surjective and continuous, already allow us to deduce where $f$ is differentiable?
And if so, how elementary can such a proof be? Yet, we don't know Brouwer fixed-point theorem etc.