Left and right derivative

In mathematics, the terms left derivative and right derivative are used to describe certain kinds of derivatives.

The most common use is to describe the one-sided limit of a function defined on a subset of the real line, (see semi-differentiability). Under this notation, the right derivative is defined as

and the left derivative as

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.

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

Proof: For a proof by contradiction, assume there exist a < b in I such that f(a) ≠ f(b). Then

Define c as the infimum of all those x in the interval (a,b] for which the difference quotient of f exceeds ε in absolute value, i.e.

Due to the continuity of f, it follows that c < b and |f(c) – f(a)| = ε(c – a). 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,

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