TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a differentiable function of one
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
variable is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
whose
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 *Doma ...
. 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 discret ...

of a differentiable function has a non-
vertical Vertical may refer to: * Vertical direction In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...

tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

at each interior point) and does not contain any break, angle, or
cusp Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative $f\text{'}\left(x_0\right)$ exists. In other words, the graph of has a non-vertical tangent line at the point .

# Differentiability of real functions of one variable

A function $f:U\subset\mathbb\to\mathbb$, defined on an open set $U$, is ''differentiable'' at $a\in U$ if the derivative :$f\text{'}\left(a\right)=\lim_\frac$ 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 ga ...
at . This function is ''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 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 ga ...
(at every point where it is differentiable). It is ''continuously differentiable'' if its derivative is also a continuous function.

# 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 ga ...
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 Cusp may refer to: Mathematics *Cusp (singularity) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, or
vertical tangent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 t ...
point. However, a result of
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

states that the set of functions that have a derivative at some point is a
meagre set In the Mathematics, mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a Set (mathematics), set that, considered as a subset of a (usually larger) topological sp ...
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 300px, Plot of Weierstrass function over the interval minus;2, 2 Like other fractals, the function exhibits self-similarity">fractal.html" ;"title="minus;2, 2 Like other fractal">minus;2, 2 Like other fractals, the function exhibi ...

.

# Differentiability classes

A function $f$ is said to be if the derivative $f^\left(x\right)$ exists and is itself a continuous function. Although the derivative of a differentiable function never has a
jump discontinuity Continuous function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical 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 of a in . Elementary rules of differentiation Unless otherwise stated, all functions are functions of that return real values; although more generally, the formul ...
imply $f'(x) = 2x\sin(1/x) - \cos(1/x)$ which has no limit as $x \to 0.$ Nevertheless,
Darboux's theorem Darboux's theorem is a theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...
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 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point ...

. Similarly to how
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are said to be of continuously differentiable functions are sometimes said to be of A function is of if the first and of the function both exist and are continuous. More generally, a function is said to be of if the first $k$ derivatives $f^\left(x\right), f^\left(x\right), \ldots, f^\left(x\right)$ 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 is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ...

or equivalently, of

# Differentiability in higher dimensions

A
function of several real variables In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
is said to be differentiable at a point if
there exists In predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a s ...
a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

such that :$\lim_ \frac = 0.$ If a function is differentiable at , then all of the
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s exist at , and the linear map is given by the
Jacobian matrix In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...
. 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 British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
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 multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...
s) does not in general guarantee that a function is differentiable at a point. For example, the function defined by :$f\left(x,y\right) = \beginx & \texty \ne x^2 \\ 0 & \texty = x^2\end$ is not differentiable at , but all of the partial derivatives and directional derivatives exist at this point. For a continuous example, the function :$f\left(x,y\right) = \beginy^3/\left(x^2+y^2\right) & \text\left(x,y\right) \ne \left(0,0\right) \\ 0 & \text\left(x,y\right) = \left(0,0\right)\end$ 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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
, 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 :$f\text{'}\left(a\right)=\lim_\frac.$ 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, $f\left(z\right)=\frac$ is differentiable at every point, viewed as the 2-variable real function $f\left(x,y\right)=x$, but it is not complex-differentiable at any point. Any function that is complex-differentiable in a neighborhood of a point is called
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
, 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''. More generally, 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'').