Semi-differentiability

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 a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left. 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 ∩ 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 ∩ 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 ∩ and I ∩ 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. 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 , at a = 0. We find easily If a function is semi-differentiable at a point a, it implies that it is continuous at a.