A question on minimizing distance function

Question

Let $C:I\to \Bbb R^2$ be a continuous and non-differentiable curve and $p=(x,y)$ a point on $\Bbb R^2$. Is the $d(p,C)$ (distance from $C$) a differentiable function? I think this function is not differentiable. but how to calculate $d(p,C)$ for given $p$?

Answer 1

If $C$ is not differentiable, then $t\mapsto dist(p,C(t))$ is in general not differentiable.

How you can find the distance from $C$ to $p$ depends on $C$. Is $C$ piecewise differentiable? If so, then you can compute the minimal distance on each piece and compare them. But don't forget the boundary for each piece!

Answer 2

Suppose $C\equiv(0,0)$. Then $d(p,C) = \|p\|$ is not differentiable. What do you mean by "minimize $d(p,C)$ for given $p$"? Is $p$ fixed?