
In
mathematics, a differentiable function of one
real
Real may refer to:
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Music Albums
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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
*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 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 whose graph is a straight line, that is, a polynomial function of degree zero or one. For di ...
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
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
. Generally speaking, is said to be of class if its first
derivatives
exist and are continuous over the domain of the function
.
Differentiability of real functions of one variable
A function
, defined on an open set
, is said to be ''differentiable'' at
if the derivative
:
exists. This implies that the function is
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
A continuous function is not necessarily differentiable, but a differentiable function is necessarily
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
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 i ...
).
Differentiability and continuity

If is differentiable at a point , then must also be
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, 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 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 that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
The Weierst ...
.
Differentiability classes

A function
is said to be if the derivative
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 ...
, it is possible for the derivative to have an
essential discontinuity. For example, the function
is differentiable at 0, since
exists. However, for
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
which 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 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 im ...
.
Similarly to how
continuous functions 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 (mathematics), function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself ...
of the function both exist and are continuous. More generally, a function is said to be of if the first
derivatives
all exist and are continuous. If derivatives
exist for all positive integers
the function is
smooth 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 functi ...
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, wh ...
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 pr ...
such that
:
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 variables ...
, 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 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
:
is not differentiable at , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function
:
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 functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
, 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
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-variable real function
, 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.
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 Fréch ...
*
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 righ ...
*
Differentiable programming
References
{{Differentiable computing
Differential calculus
Multivariable calculus
Smooth functions