differentiable function

Let $f\colon V\to W$ be a function, where $V$ and $W$ are Banach spaces. For $x\in V$ , the function $f$ is said to be differentiable at $x$ if its derivative exists at that point. Differentiability at $x\in V$ implies continuity at $x$ . If $S\subset V$ , then $f$ is said to be differentiable on $S$ if $f$ is differentiable at every point $x\in S$ .

For the most common example, a real function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable if its derivative $\frac{df}{dx}$ exists for every point in the region of interest. For another common case of a real function of $n$ variables $f(x_{1},x_{2},\ldots,x_{n})$ (more formally $f\colon\mathbb{R}^{n}\to\mathbb{R}$ ), it is not sufficient that the partial derivatives $\frac{\partial f}{\partial x_{i}}$ exist for $f$ to be differentiable. The derivative of $f$ must exist in the original sense at every point in the region of interest, where $\mathbb{R}^{n}$ is treated as a Banach space under the usual Euclidean vector norm.

If the derivative of $f$ is continuous, then $f$ is said to be $C^{1}$ . If the $k$ th derivative of $f$ is continuous, then $f$ is said to be $C^{k}$ . By convention, if $f$ is only continuous but does not have a continuous derivative, then $f$ is said to be $C^{0}$ . Note the inclusion property $C^{k+1}\subset C^{k}$ . And if the $k$ -th derivative of $f$ is continuous for all $k$ , then $f$ is said to be $C^{\infty}$ . In other words $C^{\infty}$ is the intersection $C^{\infty}=\bigcap_{k=0}^{\infty}C^{k}$ .

Differentiable functions are often referred to as smooth. If $f$ is $C^{k}$ , then $f$ is said to be $k$ -smooth. Most often a function is called smooth (without qualifiers) if $f$ is $C^{\infty}$ or $C^{1}$ , depending on the context.

Title	differentiable function
Canonical name	DifferentiableFunction
Date of creation	2013-03-22 12:39:10
Last modified on	2013-03-22 12:39:10
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	24
Author	Koro (127)
Entry type	Definition
Classification	msc 26A24
Classification	msc 57R35
Synonym	smooth function
Synonym	differentiable mapping
Synonym	differentiable map
Synonym	smooth mapping
Synonym	smooth map
Synonym	continuously differentiable
Related topic	OneSidedDerivatives
Related topic	RoundFunction
Related topic	ConverseTheorem
Related topic	WeierstrassFunction
Defines	differentiable
Defines	smooth