Incalculus, the notions ofone-sided differentiability andsemi-differentiability of areal-valuedfunctionf of a real variable are weaker thandifferentiability. Specifically, the functionf is said to beright differentiable at a pointa if, roughly speaking, aderivative can be defined as the function's argumentx moves toa from the right, andleft differentiable ata if the derivative can be defined asx moves toa from the left.

Inmathematics, aleft derivative and aright derivative arederivatives (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.

Letf denote a real-valued function defined on a subsetI of the real numbers.

Ifa ∈ I is alimit point ofI ∩ [a,∞) and theone-sided limit

${\displaystyle {\mathrm{\partial }}_{+}f(a):=\underset{\genfrac{}{}{0ex}{}{x\to a^{+}}{x\in I}}{lim}\frac{f(x)-f(a)}{x-a}}$

exists as a real number, thenf is calledright differentiable ata and the limit∂₊f(a) is called theright derivative off ata.

Ifa ∈ I is a limit point ofI ∩ (–∞,a] and the one-sided limit

${\displaystyle {\mathrm{\partial }}_{-}f(a):=\underset{\genfrac{}{}{0ex}{}{x\to a^{-}}{x\in I}}{lim}\frac{f(x)-f(a)}{x-a}}$

exists as a real number, thenf is calledleft differentiable ata and the limit∂_–f(a) is called theleft derivative off ata.

Ifa ∈ I is a limit point ofI ∩ [a,∞) andI ∩ (–∞,a] and iff is left and right differentiable ata, thenf is calledsemi-differentiable ata.

If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define asymmetric derivative, which equals thearithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.^[1]

• A function isdifferentiable at aninterior pointa of itsdomain if and only if it is semi-differentiable ata and the left derivative is equal to the right derivative.
• An example of a semi-differentiable function, which is not differentiable, is theabsolute value function${\displaystyle f(x)=|x|}$, ata = 0. We find easily${\displaystyle {\mathrm{\partial }}_{-}f(0)=-1,{\mathrm{\partial }}_{+}f(0)=1.}$
• If a function is semi-differentiable at a pointa, it implies that it is continuous ata.
• Theindicator function 1_[0,∞) is right differentiable at every reala, but discontinuous at zero (note that this indicator function is not left differentiable at zero).

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

Another common use is to describe derivatives treated asbinary operators ininfix notation, in which the derivatives is to be applied either to the left or rightoperands. This is useful, for example, when defining generalizations of thePoisson bracket. For a pair of functions f and g, the left and right derivatives are respectively defined as

${\displaystyle f\overleftarrow{{\mathrm{\partial }}_x}g=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\cdot g}$

${\displaystyle f\overrightarrow{{\mathrm{\partial }}_x}g=f\cdot \frac{\mathrm{\partial }g}{\mathrm{\partial }x}.}$

Inbra–ket notation, the derivative operator can act on the right operand as the regular derivative or on the left as the negative derivative.^[2]

This above definition can be generalized to real-valued functionsf defined on subsets ofRⁿ using a weaker version of thedirectional derivative. Leta be an interior point of the domain off. Thenf is calledsemi-differentiable at the pointa if for every directionu ∈ Rⁿ the limit

${\displaystyle {\mathrm{\partial }}_{u}f(a)=\underset{h\to 0^{+}}{lim}\frac{f(a+hu)-f(a)}{h}}$

with${\displaystyle h\in }$R exists as a real number.

Semi-differentiability is thus weaker thanGateaux differentiability, for which one takes in the limit aboveh → 0 without restrictingh to only positive values.

For example, the function${\displaystyle f(x,y)=\sqrt{x^2+y^2}}$ is semi-differentiable at${\displaystyle (0,0)}$, but not Gateaux differentiable there. Indeed,${\displaystyle f(hx,hy)=|h|f(x,y)\text{ and for }h\ge 0,f(hx,hy)=hf(x,y),f(hx,hy)/h=f(x,y),}$ with${\displaystyle a=0,u=(x,y),{\mathrm{\partial }}_{u}f(0)=f(x,y)}$

(Note that this generalization is not equivalent to the original definition forn = 1 since the concept of one-sided limit points is replaced with the stronger concept of interior points.)

• Anyconvex function on a convexopen subset ofRⁿ is semi-differentiable.
• While every semi-differentiable function of one variable is continuous; this is no longer true for several variables.

Instead of real-valued functions, one can consider functions taking values inRⁿ or in aBanach space.

• Directional derivative
• Partial derivative
• Gradient
• Gateaux derivative
• Fréchet derivative
• Derivative (generalizations)
• Phase space formulation § Star product
• Dini derivatives

• Preda, V.; Chiţescu, I. (1999). "On Constraint Qualification in Multiobjective Optimization Problems: Semidifferentiable Case".J. Optim. Theory Appl.100 (2):417–433.doi:10.1023/A:1021794505701.S2CID 119868047.

• Real analysis
• Differential calculus
• Functions and mappings

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