Class C^k function

A class $C^k$ function on an interval $I$ is a function $f:I\to\mathbb{R}$ such that $f^{(j)}$ exists on $I$ and is continuous for every integer $0\le j\le k$ (with $f^{(0)}=f$); it is class $C^\infty$ if this holds for all $k$.

This definition is built from higher derivatives together with continuity (viewed generally as a continuous map property). In several variables, the analogous notion is a class C^k map .

Examples:

• Every polynomial function is class $C^\infty$ on $\mathbb{R}$.
• The function $f(x)=|x|$ is class $C^0$ on $\mathbb{R}$ but not class $C^1$.