In calculus, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative 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 derivatives (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 a ∈ I is a limit point of I ∩ [a,∞) and the one-sided limit
∂ + f ( a ) := lim x → a + x ∈ I f ( x ) − f ( a ) x − a {\displaystyle \partial _{+}f(a):=\lim _{\scriptstyle x\to a^{+} \atop \scriptstyle x\in I}{\frac {f(x)-f(a)}{x-a}}}
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 a ∈ I is a limit point of I ∩ (–∞,a] and the one-sided limit
∂ − f ( a ) := lim x → a − x ∈ I f ( x ) − f ( a ) x − a {\displaystyle \partial _{-}f(a):=\lim _{\scriptstyle x\to a^{-} \atop \scriptstyle x\in I}{\frac {f(x)-f(a)}{x-a}}}
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 a ∈ I is a limit point of I ∩ [a,∞) and I ∩ (–∞,a] 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, which equals the arithmetic mean 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 at an interior point a of its domain 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 function f ( x ) = | x | {\displaystyle f(x)=|x|} , at a = 0. We find easily ∂ − f ( 0 ) = − 1 , ∂ + f ( 0 ) = 1. {\displaystyle \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 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 operators in infix notation, in which the derivatives is to be applied either to the left or right operands. This is useful, for example, when defining generalizations of the Poisson bracket. For a pair of functions f and g, the left and right derivatives are respectively defined as
f ∂ x ← g = ∂ f ∂ x ⋅ g {\displaystyle f{\stackrel {\leftarrow }{\partial _{x}}}g={\frac {\partial f}{\partial x}}\cdot g}
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