# 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