TITLE: Proof that no differentiable space-filling curve exists
QUESTION [26 upvotes]: 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?

REPLY [2 votes]: Using the Sard's theorem.
Assume that your curves exists. Then Each point in the domain is a critical point and so each point in the image is a critical value.
But this is a contradiction with the sard's theorem since the set of critical values has measure zero.