Smoothness

Definition

A function is smooth if it is infinitely differentiable on its entire domain. A function is smooth on some interval $(a,b)$ if it is infinitely differentiable on that interval with derivatives that remain continuous.

Differentiability is denoted as $C^k$. A function whose derivatives up to order four exist and are continuous on an interval $(a,b)$ is said to be $C^4$ (pronounced “C four”) on that interval; a function which can be differentiated infinitely many times and whose derivatives remain continuous for every order is said to be $C^{\infty }$ (“C infinity”) on (a, b)$.Thecosinefunction,forexample,is$C^{\infty}$, because when it is differentiated it transforms into the sine function, and the sine function transforms into the cosine.

$$\begin{gathered} f(x)=\cos (x) \\ {f}^{\prime }(x)=-\sin (x) \\ {f}^{\prime \prime }(x)=-\cos (x) \\ {f}^{\prime \prime }(x)=\sin (x) \\ {f}^{\prime \prime \prime }(x)=\cos (x) \\ \dots \\ \therefore f(x)\in C^{\infty }(\mathbb{R}) \end{gathered}$$

A smooth function must be $C^{\infty }$.