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Proof that no differentiable space-filling curve exists

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?

This is answered for $C^1$ curves here, and then the $C^1$ condition is weakened here.

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.

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Proof that no differentiable space-filling curve exists

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?

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This is answered for $C^1$ curves here, and then the $C^1$ condition is weakened here.

@GeorgeLowther's proof (your 1st link, in response to an MSE question, "Range of curve on a compact interval is nowhere dense"), is very clean! - Joseph O'Rourke Mar 30 at 0:33

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