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单词 DifferentiableFunction
释义

differentiable function


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

For the most common example, a real function f: is differentiableif its derivative dfdx exists for every point in the region ofinterest. For another common case of a real function of n variablesf(x1,x2,,xn) (more formally f:n),it is not sufficient that the partial derivativesMathworldPlanetmathfxi exist for f to be differentiable. Thederivative of f must exist in the original senseat every point in the region of interest,where n is treated as a Banach space under the usual Euclidean vectornorm.

If the derivative of f is continuousMathworldPlanetmathPlanetmath, then f is said to be C1. Ifthe kth derivative of f is continuous, then f is said to be Ck. By convention, if fis only continuous but does not have a continuous derivative, then f is said tobe 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 theintersectionMathworldPlanetmath C=k=0Ck.

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

Titledifferentiable function
Canonical nameDifferentiableFunction
Date of creation2013-03-22 12:39:10
Last modified on2013-03-22 12:39:10
OwnerKoro (127)
Last modified byKoro (127)
Numerical id24
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 26A24
Classificationmsc 57R35
Synonymsmooth function
Synonymdifferentiable mapping
Synonymdifferentiable map
Synonymsmooth mapping
Synonymsmooth map
Synonymcontinuously differentiable
Related topicOneSidedDerivatives
Related topicRoundFunction
Related topicConverseTheorem
Related topicWeierstrassFunction
Definesdifferentiable
Definessmooth
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更新时间:2025/5/5 7:49:43