Sometimes, in order to avoid local optimization it is a good idea to smooth a function f(x) with a kernel K(x), i.e.,
$$F(x) = \int K(x-t) \, f(t) \, \mathrm{d}t$$
I know some of the conditions for the function $K(x)$ to work, like its integral $\int K(t)\, \mathrm{d}t$ must exist, but what kind of functions works in general? For example, would it work if the function $K(x)$ is discontinuous or non-differentiable? And why?