One-sided derivative

In calculus, a branch of mathematics, 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 the right (increases) from the value a, and left differentiable at a if the derivative can be defined as x moves to the left from a.

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.

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

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

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.

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