Hmm...
The Heaviside function,
H(x) = \int_{-\infty}^x \delta(t) \: dt
shows what you mean, but how is it that many people define H(0) = 1/2 ? I think I'm missing the point...
If I had a function g(x) defined by
g(x) = \int_{-\infty}^{\infty} f(x) \delta(x) dx
where \delta(x) is the dirac delta function, what would dg(x)/dx be? The fundamental theorem of calculus requires that f(x) \delta(x) needs to be a continuous and differentiable function before I can...