differentiable function

Let f:VW be a function, where V and W are Banach spacesMathworldPlanetmath. For xV, the function f is said to be differentiableMathworldPlanetmathPlanetmath at x if its derivativePlanetmathPlanetmath exists at that point. Differentiability at xV implies continuity at x. If SV, then f is said to be differentiable on S if f is differentiable at every point xS.

For the most common example, a real function f: is differentiable if its derivative dfdx exists for every point in the region of interest. For another common case of a real function of n variables f(x1,x2,,xn) (more formally f:n), it is not sufficient that the partial derivativesMathworldPlanetmath fxi 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 n is treated as a Banach space under the usual Euclidean vector norm.

If the derivative of f is continuousMathworldPlanetmathPlanetmath, then f is said to be C1. If the kth derivative of f is continuous, then f is said to be Ck. By convention, if f is only continuous but does not have a continuous derivative, then f is said to be C0. Note the inclusion property Ck+1Ck. And if the k-th derivative of f is continuous for all k, then f is said to be C. In other words C is the intersectionMathworldPlanetmath C=k=0Ck.

Differentiable functions are often referred to as smooth. If f is Ck, then f is said to be k-smooth. Most often a function is called smooth (without qualifiers) if f is C or C1, 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