User blog comment:Nayuta Ito/generalization of arrow notation/@comment-11227630-20141026002048

A generalization to reals should have this property: f(x) is infinitely differentiable for any x. Your extension doesn't fit that.

An example is the generalization of factorial. There may be tons of generalizations.

A simple one is
 * x!=1 iff 0<x<=1
 * (x+1)!=x!*(x+1)

But it's not infinitely differentiable for any x.

Another generalization is the Gamma function, which is infinitely differentiable for any x.