Semi-differentiability

One-dimensional case

 

Let $latx f$ denote a real-valued function defined on a subset I of the real numbers \mathbf{R}. If a\in I is a limit point of I\cap [a,+\infty) and the one-side limit

\displaystyle f_+(a):=\lim_{x\to a+: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 limt f_+(a) is called the right derivative of f at a.

Similarily, if a\in I is a limit point of I\cap (-\infty,a] and the one-side limit

\displaystyle f_-(a):=\lim_{x\to a-,x\in }\frac{f(x)-f(a)}{x-a}

exists as a real number, then f is called left derivative at a and the limit f_-(a) is called the left derivative of f at a.

If a\in I is a limit point of I\cap [a,+\infty) and I\cap (-\infty,a] and if f is left and right differentiable at a, then f is called semi-differentiable at a.

 

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