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 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 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
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
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
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 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.
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, 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 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.
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 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 is said to be differentiable at a point if
there exists a
linear map such that
:
If a function is differentiable at , then all of the
partial derivatives exist at , and the linear map is given by the
Jacobian matrix, 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 derivatives) 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-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
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, 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