Smoothness classes

Remarks

The classes are nested: $C^\infty(I) \subseteq \cdots \subseteq C^2(I) \subseteq C^1(I) \subseteq C^0(I)$. The class $C^1$ is the class of continuously differentiable functions, and $C^\infty$ is the class of smooth functions. The class $C^\omega$ of real analytic functions lies inside $C^\infty$. For functions of several variables, $C^k$ is defined by continuous partial derivatives up to order $k$; see the characterisation by derivatives.

Example: $f(x) = x \abs{x}$ on $\R$ is of class $C^1$. Its derivative is $f'(x) = 2 \abs{x}$, which is continuous. It is not of class $C^2$, because $2\abs{x}$ is not differentiable at $0$. More generally, $x^k \abs{x}$ is of class $C^k$ but not of class $C^{k+1}$.

Non-example: $g(x) = x^2 \sin(1/x)$ for $x \ne 0$, with $g(0) = 0$, is differentiable on $\R$ but not of class $C^1$. Its derivative is $g'(x) = 2x \sin(1/x) - \cos(1/x)$ for $x \ne 0$ and $g'(0) = 0$, and $g'$ has no limit at $0$.