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.

Let f denote a real-valued function defined on a subset I of the real numbers.

If $a \in I$ is a limit point of $I \cap$ $[a,\infty)$ 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 $a \in I$ is a limit point of $I \cap$ $(-\infty, a]$ 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 $a \in I$ 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.

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.

Theorem — Let f be a real-valued, continuous function, defined on an arbitrary interval I of the real line. If f is right differentiable at every point $a \in I$ , which is not the supremum of the interval, and if this right derivative is always zero, then f is constant.

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