 Smooth function

A smooth function has continuous derivatives of every finite order, usually denoted $C^{\infty }$. For a function on an open subset of Euclidean space, this means every finite-order partial derivative exists and is continuous.

Every polynomial is smooth. Smoothness is weaker than being real analytic: the function $f(x)=e^{-1/x^2}$ for $x\ne 0$, extended by $f(0)=0$, is smooth and has every derivative zero at zero, but is positive elsewhere. Its Taylor series at zero therefore does not reproduce it on any neighborhood. The distinction matters when an argument requires convergence of a power-series representation rather than merely repeated differentiation.