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In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued function ''f'' of a real variable are weaker than
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in i ...
. Specifically, the function ''f'' is said to be right differentiable at a point ''a'' if, roughly speaking, a
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. ...
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, a left derivative and a right derivative are
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. ...
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, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
of   and the
one-sided limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches ...
:\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 operation generalizing the ordinary derivative. It is defined asThomson, p. 1. : \lim_ \frac. The expression under the limit is sometimes called the symmetric difference quotient. A function is sai ...
, which equals the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The coll ...
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 mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
at an
interior point In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
''a'' of 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 * ...
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, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function f(x)=, x, , at ''a'' = 0. We find easily \partial_-f(0)=-1, \partial_+f(0)=1. * If a function is semi-differentiable at a point ''a'', it implies that it is continuous at ''a''. * The
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
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, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary o ...
s in
infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in . Usage Binary relations are ...
, 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 examp ...
s. This is useful, for example, when defining generalizations of the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
. 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, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathemat ...
, 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 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 ...
. 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 with h \in R 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. Indeed, f(hx,hy)=, h, f(x,y) \text h \geq 0, f(hx,hy)=h f(x,y), f(hx,hy)/h=f(x,y), with a= 0, u=(x,y), \partial_uf(0)=f(x,y) (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 on a convex
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
.


See also

*
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. ...
*
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 ...
*
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). Par ...
*
Gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
*
Gateaux derivative In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined ...
*
Fréchet derivative In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...
*
Derivative (generalizations) 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 ...
* Phase space formulation#Star product * Dini derivatives


References

* {{cite journal , last1=Preda , first1=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 , s2cid=119868047 Real analysis Differential calculus Articles containing proofs Functions and mappings