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In
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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
variable is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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. F ...
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 * Do ...
. In other words, the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of a differentiable function has a non-
vertical Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
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. More ...
at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
at each interior point) and does not contain any break, angle, or
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
. 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 ...
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 ...
(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).


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 ...
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 A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
, or
vertical tangent In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency. Limit definiti ...
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 t ...
point. However, a result of
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...
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 (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve ...
.


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 ...
, it is possible for the derivative to have an
essential 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 ...
. 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 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 Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
implies that the derivative of any function satisfies the conclusion of the intermediate value theorem. 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 value ...
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 In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
or equivalently, of


Differentiability in higher dimensions

A
function of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
is said to be differentiable at a point if
there exists In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
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 Map (mathematics), mapping V \to W between two vect ...
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). Part ...
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 variables as ...
, an ''n'' × ''m'' matrix in this case. A similar formulation of the higher-dimensional derivative is provided by the
fundamental increment lemma In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative ''f''(''a'') of a function ''f'' at a point ''a'': :f'(a) = \lim_ \frac. The lemma asserts that the existence o ...
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 ...
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 In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
at that point. Such a function is necessarily infinitely differentiable, and in fact analytic.


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 In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet ...
*
Semi-differentiability In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function ''f'' of a real variable are weaker than differentiability. Specifically, the function ''f'' is said to be right ...
*
Differentiable programming Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation. This allows for gradient-based optimization of parameters in the program, often via grad ...


References

{{Differentiable computing Differential calculus Multivariable calculus Smooth functions