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In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, a branch of
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 ...
, the notions of one-sided differentiability and semi-differentiability of a
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 ...
-valued
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 ...
''f'' of a real variable are weaker than
differentiability In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...
. Specifically, the function ''f'' is said to be right differentiable at a point ''a'' if, roughly speaking, a
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 ...
can be defined as the function's argument ''x'' moves to ''a'' from the right, and left differentiable at ''a'' if the derivative can be defined as ''x'' moves to ''a'' from the left.


One-dimensional case

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 left derivative and a right derivative are
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
s (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.


Definitions

Let ''f'' denote a real-valued function defined on a subset ''I'' of the real numbers. If is a
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
of   and the
one-sided limit In calculus, a one-sided limit is either of the two Limit of a function, limits of a function (mathematics), function ''f''(''x'') of a real number, real variable ''x'' as ''x'' approaches a specified point either from the left or from the right. ...
:\partial_+f(a):=\lim_\frac exists as a real number, then ''f'' is called right differentiable at ''a'' and the limit ''∂''+''f''(''a'') is called the right derivative of ''f'' at ''a''. If is a limit point of   and the one-sided limit :\partial_-f(a):=\lim_\frac exists as a real number, then ''f'' is called left differentiable at ''a'' and the limit ''∂''''f''(''a'') is called the left derivative of ''f'' at ''a''. If is a limit point of   and and if ''f'' is left and right differentiable at ''a'', then ''f'' is called semi-differentiable at ''a''. If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a
symmetric derivative In mathematics, the symmetric derivative is an Operator (mathematics), operation generalizing the ordinary derivative. It is defined asThomson, p. 1. : \lim_ \frac. The expression under the limit is sometimes called the symmetric difference quoti ...
, which equals the
arithmetic mean 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 ...
of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.


Remarks and examples

* A function is
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
at an
interior point In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...

interior point
''a'' of 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 ...
if and only if it is semi-differentiable at ''a'' and the left derivative is equal to the right derivative. * An example of a semi-differentiable function, which is not differentiable, is the
absolute value 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 an ...

absolute value
at ''a'' = 0. * If a function is semi-differentiable at a point ''a'', it implies that it is continuous at ''a''. * The
indicator 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 a ...
1[0,∞) is right differentiable at every real ''a'', but discontinuous at zero (note that this indicator function is not left differentiable at zero).


Application

If a real-valued, differentiable function ''f'', defined on an interval ''I'' of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of ''f''. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.


Differential operators acting to the left or the right

Another common use is to describe derivatives treated as
binary operator 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). I ...
s in
infix notation Infix notation is the notation commonly used in arithmetical and logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ) ...
, in which the derivatives is to be applied either to the left or right
operand In mathematics an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above example ...
s. This is useful, for example, when defining generalizations of the
Poisson bracket 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 ...
. For a pair of functions f and g, the left and right derivatives are respectively defined as :f \stackrel_x g = \frac \cdot g :f \stackrel_x g = f \cdot \frac. In
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum ...
, the derivative operator can act on the right operand as the regular derivative or on the left as the negative derivative.


Higher-dimensional case

This above definition can be generalized to real-valued functions ''f'' defined on subsets of R''n'' using a weaker version of 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 ...
. Let ''a'' be an interior point of the domain of ''f''. Then ''f'' is called ''semi-differentiable'' at the point ''a'' if for every direction ''u'' ∈ R''n'' the limit :\partial_uf(a)=\lim_\frac exists as a real number. Semi-differentiability is thus weaker than Gateaux differentiability, for which one takes in the limit above ''h'' → 0 without restricting ''h'' to only positive values. For example, the function f(x, y) = \sqrt is semi-differentiable at (0, 0), but not Gateaux differentiable there. (Note that this generalization is not equivalent to the original definition for ''n = 1'' since the concept of one-sided limit points is replaced with the stronger concept of interior points.)


Properties

* Any
convex function (in green) is a convex set File:Convex polygon illustration2.svg, Illustration of a non-convex set. Illustrated by the above line segment whereby it changes from the black color to the red color. Exemplifying why this above set, illustrated in gr ...

convex function
on a convex
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of R''n'' is semi-differentiable. * While every semi-differentiable function of one variable is continuous; this is no longer true for several variables.


Generalization

Instead of real-valued functions, one can consider functions taking values in R''n'' or in a
Banach space 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 th ...
.


See also

*
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
*
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 ...
*
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 ...

Partial derivative
*
Gradient 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 * Prod ...

Gradient
*
Gateaux derivative 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 h ...
*
Fréchet 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 ...
*
Derivative (generalizations) 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 ...
* Phase space formulation#Star product * Dini derivatives


References

* {{cite journal , last=Preda , first=V. , last2=Chiţescu , first2=I. , title=On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case , journal=J. Optim. Theory Appl. , volume=100 , year=1999 , issue=2 , pages=417–433 , doi=10.1023/A:1021794505701 Real analysis Differential calculus Articles containing proofs Functions and mappings