Somebody asked about this but that thread was closed very soon. In physics, discontinuous paths breaks locality so they must be 0; but mathematically, they causes some problems. Discontinuous functions must not be differentiable, so it's impossible to calculate the action over that path. However this does not say that one will be zero, it will be infinity. This really causes a lot of problems while evaluating path integrals. Can anyone explain more? Another problem: path integral formula allows to take some ugly functions like Weierstrass function into the integration. It's not differentiable but there seems no reason to exclude these functions.