Does Euler-lagrange Equation hold with discontinuous integrand?

Question

Suppose I have a function $Heaviside(x-1) \sqrt{1+f'(x)^2}$ or some other discontinuous function, how do I find the minima with E-L Eqn or in other way?

Answer 1

For this example, the discontinuity is just a boundary between two regions where Euler-Lagrange, or other methods, can be applied. So examine the regions separately, examine the function at the boundary (limits from either side), and compare the points. The minima will lie among all of them.

It is only when the function is not differentiable at more than isolated points that differential methods fail completely.