I've defined it as $D_{i}f(x) = lim_{h \rightarrow 0} h(f(x+h)+f(x))$. What do you think /sci/?
Polynomials are always 0 but curiously, some non differentiable undefined spots in a good handful of functions are idiot differentiable. For example, let $g(x) = \frac{1}{x}$, then $D_{i} g(h) = 2$.