# Smooth function

Notice that any additive function $f$ (i.e. $f(x+y)=f(x)+f(y)$ for all $x$ and $y$) is smooth. There exist additive functions that are continuous at no point.
The notion of a smooth function as introduced above is a rather uncommon one. Usually "smooth function" means "sufficient often differentiable function", most often $C^\infty$-function (infinitely often differentiable function); it can also mean "having modulus of smoothness satisfying certain growth conditions" (cf. also Smoothness, modulus of).