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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a differentiable function of one real variable is a function whose
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
exists at each point in its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
. In other words, the graph of a differentiable function has a non- vertical
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the function f.


Differentiability of real functions of one variable

A function f:U\to\mathbb, defined on an open set U\subset\mathbb, is said to be ''differentiable'' at a\in U if the derivative :f'(a)=\lim_\frac exists. This implies that the function is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
at . This function is said to be ''differentiable'' on if it is differentiable at every point of . In this case, the derivative of is thus a function from into \mathbb R. A continuous function is not necessarily differentiable, but a differentiable function is necessarily
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
(at every point where it is differentiable) as being shown below (in the section Differentiability and continuity). A function is said to be ''continuously differentiable'' if its derivative is also a continuous function; there exists a function that is differentiable but not continuously differentiable as being shown below (in the section
Differentiability classes In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
).


Differentiability and continuity

If is differentiable at a point , then must also be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
at . 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, or vertical 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 at
almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a
meagre set In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
in the space of all continuous functions.. Cited by 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 the
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstr ...
.


Differentiability classes

A function f is said to be if the derivative f^(x) exists and is itself a continuous function. Although the derivative of a differentiable function never has a
jump discontinuity Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...
, it is possible for the derivative to have an essential discontinuity. For example, the function f(x) \;=\; \begin x^2 \sin(1/x) & \textx \neq 0 \\ 0 & \text x = 0\end is differentiable at 0, since f'(0) = \lim_ \left(\frac\right) = 0 exists. However, for x \neq 0,
differentiation rules This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that return real ...
imply f'(x) = 2x\sin(1/x) - \cos(1/x)\;, which has no limit as x \to 0. 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 the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...
. Similarly to how
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s are said to be of continuously differentiable functions are sometimes said to be of A function is of if the first and second derivative of the function both exist and are continuous. More generally, a function is said to be of if the first k derivatives f^(x), f^(x), \ldots, f^(x) all exist and are continuous. If derivatives f^ exist for all positive integers n, the function is smooth or equivalently, of


Differentiability in higher dimensions

A function of several real variables is said to be differentiable at a point if there exists a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
such that :\lim_ \frac = 0. If a function is differentiable at , then all of the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s exist at , and the linear map is given by the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
, an ''n'' × ''m'' matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the fundamental increment lemma found in single-variable calculus. If all the partial derivatives of a function exist in a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of a point and are continuous at the point , then the function is differentiable at that point . However, the existence of the partial derivatives (or even of all the
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
s) does not guarantee that a function is differentiable at a point. For example, the function defined by :f(x,y) = \beginx & \texty \ne x^2 \\ 0 & \texty = x^2\end is not differentiable at , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function :f(x,y) = \beginy^3/(x^2+y^2) & \text(x,y) \ne (0,0) \\ 0 & \text(x,y) = (0,0)\end is not differentiable at , but again all of the partial derivatives and directional derivatives exist.


Differentiability in complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers. So, a function f:\mathbb\to\mathbb is said to be differentiable at x=a when :f'(a)=\lim_\frac. Although this definition looks similar to the differentiability of single-variable real functions, it is however a more restrictive condition. A function f:\mathbb\to\mathbb, that is complex-differentiable at a point x=a is automatically differentiable at that point, when viewed as a function f:\mathbb^2\to\mathbb^2. This is because the complex-differentiability implies that :\lim_\frac=0. However, a function f:\mathbb\to\mathbb can be differentiable as a multi-variable function, while not being complex-differentiable. For example, f(z)=\frac is differentiable at every point, viewed as the 2-variable real function f(x,y)=x, but it is not complex-differentiable at any point. Any function that is complex-differentiable in a neighborhood of a point is called holomorphic at that point. Such a function is necessarily infinitely differentiable, and in fact
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
.


Differentiable functions on manifolds

If ''M'' is a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, a real or complex-valued function ''f'' on ''M'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate chart defined around ''p''. If ''M'' and ''N'' are differentiable manifolds, a function ''f'': ''M'' → ''N'' is said to be differentiable at a point ''p'' if it is differentiable with respect to some (or any) coordinate charts defined around ''p'' and ''f''(''p'').


See also

* Generalizations of the derivative * Semi-differentiability * Differentiable programming


References

{{Differentiable computing Differential calculus Multivariable calculus Smooth functions