




Jump discontinuities only cause trouble is you try to differentiate the function at the jump. It is possible to get around even with this difficulty by considering as , with and defining its derivative to be 2d(x), .
This object d(x) is called a "delta function". So defined, it is not really a function, since it is 0 except at x = 0, where it is infinite.It is called a "distribution" and is occasionally useful.
Physically it corresponds to density function of a point in one dimension.