Inmathematics, adifferentiable function of onereal variable is afunction whosederivative exists at each point in itsdomain. In other words, thegraph of a differentiable function has a non-verticaltangent line at each interior point in its domain. A differentiable function issmooth (the function is locally well approximated as alinear function at each interior point) and does not contain any break, angle, orcusp.
Ifx0 is an interior point in the domain of a functionf, thenf is said to bedifferentiable atx0 if the derivative exists. In other words, the graph off has a non-vertical tangent line at the point(x0,f(x0)).f is said to be differentiable onU if it is differentiable at every point ofU.f is said to becontinuously differentiable if its derivative is also a continuous function over the domain of the function. Generally speaking,f is said to be of class if its first derivatives exist and are continuous over the domain of the function.
For a multivariable function, as shownhere, the differentiability of it is something more complex than the existence of the partial derivatives of it.
A function, defined on an open set, is said to bedifferentiable at if the derivative
exists. This implies that the function iscontinuous ata.
This functionf is said to bedifferentiable onU if it is differentiable at every point ofU. In this case, the derivative off is thus a function fromU into
A continuous function is not necessarily differentiable, but a differentiable function is necessarilycontinuous (at every point where it is differentiable) as is shown below (in the sectionDifferentiability and continuity). A function is said to becontinuously differentiable if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the sectionDifferentiability classes).
The above definition can be extended to define the derivative atboundary points. The derivative of a function defined on a closed subset of the real numbers, evaluated at a boundary point, can be defined as the following one-sided limit, where the argument approaches such that it is always within:
For to remain within, which is a subset of the reals, it follows that this limit will be defined as either
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Iff is differentiable at a pointx0, thenf must also becontinuous atx0. In particular, any differentiable function must be continuous at every point in its domain.The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend,cusp, orvertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
Most functions that occur in practice have derivatives at all points or atalmost every point. However, a result ofStefan Banach states that the set of functions that have a derivative at some point is ameagre set in the space of all continuous functions.[1] Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is theWeierstrass function.
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A function is said to becontinuously differentiable if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has ajump discontinuity, it is possible for the derivative to have anessential discontinuity. For example, the functionis differentiable at 0, sinceexists. However, fordifferentiation rules implywhich has no limit as Thus, this example shows the existence of a function that is differentiable but not continuously differentiable (i.e., the derivative is not a continuous function). Nevertheless,Darboux's theorem implies that the derivative of any function satisfies the conclusion of theintermediate value theorem.
Similarly to howcontinuous functions are said to be ofclass continuously differentiable functions are sometimes said to be ofclass. A function is ofclass if the first andsecond derivative of the function both exist and are continuous. More generally, a function is said to be ofclass if the first derivatives all exist and are continuous. If derivatives exist for all positive integers the function issmooth or equivalently, ofclass
Afunction of several real variablesf:Rm →Rn is said to be differentiable at a pointx0 ifthere exists alinear mapJ:Rm →Rn such that
If a function is differentiable atx0, then all of thepartial derivatives exist atx0, and the linear mapJ is given by theJacobian matrix, ann ×m matrix in this case. A similar formulation of the higher-dimensional derivative is provided by thefundamental increment lemma found in single-variable calculus.
If all the partial derivatives of a function exist in aneighborhood of a pointx0 and are continuous at the pointx0, then the function is differentiable at that pointx0.
However, the existence of the partial derivatives (or even of all thedirectional derivatives) does not guarantee that a function is differentiable at a point. For example, the functionf:R2 →R defined by
is not differentiable at(0, 0), but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function
is not differentiable at(0, 0), but again all of the partial derivatives and directional derivatives exist.
Incomplex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividingcomplex numbers. So, a function is said to be differentiable at when
Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function, that is complex-differentiable at a point is automatically differentiable at that point, when viewed as a function. This is because the complex-differentiability implies that
However, a function can be differentiable as a multi-variable function, while not being complex-differentiable. For example, is differentiable at every point, viewed as the 2-variablereal function, but it is not complex-differentiable at any point because the limit gives different values for different approaches to 0.
Any function that is complex-differentiable in a neighborhood of a point is calledholomorphic at that point. Such a function is necessarily infinitely differentiable, and in factanalytic.
IfM is adifferentiable manifold, a real or complex-valued functionf onM is said to be differentiable at a pointp if it is differentiable with respect to some (or any) coordinate chart defined aroundp. IfM andN are differentiable manifolds, a functionf: M → N is said to be differentiable at a pointp if it is differentiable with respect to some (or any) coordinate charts defined aroundp andf(p).
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