Differentiability Class¶

Definition¶

Let $X\subseteq \mathbb{R}$ be an open interval and let $k\geq0$ be an integer. A function $f\colon X\to\mathbb{R}$ is said to be in differentiability class $C^k(X)$ if $f^{(i)}$ exists and is continuous on $X$ for every $i=0,\ldots,k$$0,\dots ,k$, where $f^{(0)}=f$$f$. A function $f\colon X\to\mathbb{R}$ is said to be infinitely differentiable, smooth, or in class $C^\infty(X)\text{,}$ if it has derivatives of all orders. On a closed interval, $C^k$ means that the interior derivatives extend continuously to both endpoints.

Interpretation¶

The differentiability class measures how smooth a function is: $f\in C^k$ means that $f$ can be differentiated $k$ times and that $f$ and all these derivatives are continuous. A larger $k$ is a stronger requirement, so $C^\infty\subseteq\cdots\subseteq C^2\subseteq C^1\subseteq C^0$$C^0$.

The three graphs climb this ladder at $0$. The function $|x|$ is continuous but has a corner, so it is in $C^0$ but not in $C^1$. The function $x|x|$ is differentiable with derivative $2|x|$ (dashed, shifted down), which is continuous but has a corner, so $x|x|$ is in $C^1$ but not in $C^2$. The cubic $x^3$ has derivatives $3x^2$, $6x$, $6$, $0,\ldots$ of all orders and is in $C^\infty$.

The continuity of the derivatives is a real condition, not a formality. The function $f(x)=x^2\sin(1/x)$ with $f(0)=0$ is differentiable everywhere, with $f'(0)=0$$0$, but $f'(x)=2x\sin(1/x)-\cos(1/x)$ for $x\neq0$ has no limit as $x\to0$$0$. So $f$ is differentiable on $\mathbb{R}$ without being in $C^1(\mathbb{R})$. In the same way, a twice differentiable function need not be in $C^2$.

Classes are the usual way to state how much smoothness a theorem needs. Math IV, for instance, works with twice continuously differentiable functions of several variables.

Key Results¶

• A function in class $C^k$ has continuous derivatives up to order $k\text{.}$
• The class $C^0$ contains all continuous functions.
• The class $C^{k+1}$ is contained in $C^k\text{.}$
• Smooth functions belong to all differentiability classes.

Examples¶

• Polynomial functions belong to $C^\infty(\mathbb{R})\text{.}$
• The function $f(x)=|x|$ belongs to $C^0(\mathbb{R})$ but not to $C^1(\mathbb{R})\text{.}$

Prerequisites¶

• Higher-Order Derivative
• Continuity

Related Concepts¶

• Real Analytic Function
• Symmetry of Second Derivatives
• Twice Continuous Differentiability on $\mathbb{R}^n$

Sources¶

• Math I: Calculus · Lecture 9

Aliases: smoothness class Keywords: continuously differentiable function, smooth function Symbols: element of, infinity, real numbers, subset

Last updated: 20 October 2026 03:02