Could someone provide a reference or a sketch of a proof that no differentiable space-filling curve exists?
Or piecewise differentiable?
Must every continuous space-filling curve be nowhere differentiable?



Proof that no differentiable space-filling curve exists
The image of an interval under a Lipschitz map has finite $1$-dimensional Hausdorff measure.


EDIT: Here's a corrected version of Pablo Shmerkin's construction. Suppose $f: \mathbb R \to \mathbb R^d$ is differentiable.


For positive integers $m,n$ let $A\_{m,n} = \{x: |y -x| \le 1/n \implies \|f(y) - f(x)\| \le m |y - x|\}$. 
For $k \in \mathbb Z$ let $A\_{m,n,k} = A\_{m,n} \cap [(k-1)/n, k/n]$. Then 
 $\bigcup\_{m,n,k} A\_{m,n,k} = \mathbb R$, and $f$ is Lipschitz on $A\_{m,n,k}$ with Lipschitz constant $m$. 


We conclude that $f(\mathbb R)$ has $\sigma$-finite $1$-dimensional Hausdorff measure, which in particular implies that it has $2$-dimensional Lebesgue measure $0$.


