Just out of pure curiosity ...
Suppose I want to connect the two points $(0,0)$ and $(1,1)$ with the graph of some continuous and differentiable function
$$f : [0; 1] \to [0; 1]$$
and let $s$ be the arc length of that function in $[0; 1]$.
Of course, the function with minimum $s$ that satisfies the above conditions is $f(x) = x$ with $s = \sqrt 2$. So for $s = \sqrt 2$, exactly one matching function can be found.
But what happens to the number of these functions if $s$ increases?
Surely, more functions can be found to match the given arc length - uncountably many more I suppose due to the nature of the real numbers.
But intuitively, I'd think that the number of such functions grows even more the greater $s$ gets, since there is more "space" the graph can use.
So, despite continuum cardinality, are there any means of measuring the number of such functions against $s$ or is it all the same once that minimal way of $f(x) = x$ as been taken?
And would this change if we limited the ways of constructing such functions to e.g. some elementary ones?