Semi-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.

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 I ∩ (–∞,a] and if f is left and right differentiable at a, then f is called semi-differentiable at a.

[1] 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.

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 ∈ I, which is not the supremum of the interval, and if this right derivative is always zero, then f is constant.

At c the right derivative of f is zero by assumption, hence there exists d in the interval (c,b] with |f(x) – f(c)| ≤ ε(x – c) for all x in (c,d].

Hence, by the triangle inequality, for all x in [c,d), which contradicts the definition of c. 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.

[2] This above definition can be generalized to real-valued functions f defined on subsets of Rn using a weaker version of the directional derivative.

Semi-differentiability is thus weaker than Gateaux differentiability, for which one takes in the limit above h → 0 without restricting h to only positive values.

(Note that this generalization is not equivalent to the original definition for n = 1 since the concept of one-sided limit points is replaced with the stronger concept of interior points.)