To be more precise I am interested in questions similar to the one below (I asked the question below on math.stackexchange last week but got not answer.)
I have a $C^1$ function $f:[0,1]^2 \to \mathbb{R}$. The arclength of the graph of $f_y:x\mapsto f(x,y)$ is $$L(y)=\int_0^1 \sqrt{1+\left(\frac{\partial f}{\partial x}(x,y)\right)^2} dx$$ Deriving under the integral does not seem to be possible since it would involve a crossed derivative $\partial^2 f/\partial x\partial y$ whose existence is not assumed. Hence I do not expect L to always be a $C^1$ function of y. But so far I have not been able to find a counterexample. This question must have been already looked at. Any idea?