Continuously differentiable function

Definition

Let $I$ be an open interval and let $f \colon I \to \mathbb{R}$. The function $f$ is continuously differentiable, or of class $C^1$, if $f$ is differentiable on $I$ and $f'$ is continuous on $I$. More generally, for $k \ge 1$, $f$ is of class $C^k$ if its higher derivative $f^{(k)}$ exists and is continuous on $I$. The class $C^0$ consists of the continuous functions.

Remarks

Each $C^{k+1}$ function is $C^k$. A differentiable function need not be $C^1$. See x^2 sin(1/x). The function $x |x|$ is $C^1$ but not twice differentiable at $0$. Mathlib defines the class as ContDiff ℝ k f.

Explanations

L1 human

A $C^1$ function has a slope at each point, and the slope changes without jumps. The class $C^k$ asks the same of the first $k$ derivatives.