Smoothness

Smoothness describes the degree of differentiability of a function. The class $C^k$ has continuous derivatives through order $k$; a smooth function is of class $C^{\infty }$.

A smooth function has continuous derivatives of every order.

A smooth function is flat at $a$ when every derivative, including order zero, vanishes at $a$. The function $e^{-1/x^2}$ off zero, extended by zero at zero, is flat there but not identically zero near zero. Thus its Taylor series does not determine it. A real analytic function that is flat at a point vanishes on a neighborhood of that point.

Extend this expression by zero at zero. Gaussian decay dominates every power, so the resulting function and its derivative difference quotients vanish to every order at zero. Away from zero, differentiation replaces $p(t)$ by $2t^{3}p(t)-t^2{p}^{\prime }(t)$, another polynomial. Induction therefore gives a smooth function flat at zero. If $p$ is nonzero, its Taylor series is zero while the function is nonzero at every sufficiently small nonzero argument, so it is not real analytic there.

A smooth cutoff function is a smooth function used to localize an argument, typically equal to one on one region and zero outside a slightly larger region. Such functions can be built from bump functions and smooth partitions of unity.

A bump function is a smooth function with compact support. Rescaled and translated bump functions provide local cutoffs.