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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 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 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 ...

derivative
exists at each point in its
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
. 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 ...

graph
of a differentiable function has a non-
vertical Vertical 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 straight lines that form an "X" * Inter ...

vertical
tangent line In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent line
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 (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

linear function
at each interior point) and does not contain any break, angle, or Cusp (singularity), 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 .


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'(a)=\lim_\frac exists. This implies that the function is continuous function, continuous 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 function, continuous (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 function, 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 (singularity), 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 everywhere, almost every 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 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, 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 imply f'(x) = 2x\sin(1/x) - \cos(1/x) which has no limit as x \to 0. Nevertheless, Darboux's theorem (analysis), 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 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 function, 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 :\lim_ \frac = 0. If a function is differentiable at , then all of the partial derivatives exist at , and the linear map is given by the Jacobian matrix. 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 Neighbourhood (mathematics), 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 in general 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-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 function, holomorphic at that point. Such a function is necessarily infinitely differentiable, and in fact Analytic function, analytic.


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''. 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'').


See also

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


References

{{Differentiable computing Differential calculus Multivariable calculus Smooth functions