Generalized one-sided derivatives

Question

Definition:Let f be a real valued function with right and left hand limits $f(x^{+} )$ and $f(x^{-})$ respectively at $x$. Then the generalized right and left hand derivatives off at x are respectively $$f'_{R}(x)=\lim_{t\to x^{+}}\frac{f(t)-f(x^{+})}{t-x}$$ $$f'_{L}(x)=\lim_{t\to x^{-}}\frac{f(t)-f(x^{-})}{t-x}$$ What is the motivation of this definition? Where can I find more information about generalized one sided derivatives?.

Answer 1

Here is one suggestion:

A function $f:\mathbb R\to\mathbb R$ is said to be regulated if one-side limits exist at every point. Such functions play an important role in many investigations, where continuity would be assuming too much. Regulated functions can be described as well as the uniform limits of step functions. All monotone functions are regulated. Regulated functions are Riemann integrable on any compact interval.

If you were interested in derivatives of regulated functions then there is a countable set of points at which the function might be discontinuous. If $f$ is discontinuous at a point $x_0$ then the actual value $f(x_0)$ won't be of much interest. Also at a discontinuity $f(x_0+)$ and $f(x_0-)$ might be different.

So, if derivatives are what you want at those discontinuities, it is pretty obvious that the definition that you are going to use would be:

$$f'_{+}(x_0)=\lim_{t\to x_0+}\frac{f(t)-f(x_0+)}{t-x_0}$$ and $$f'_{-}(x_0)=\lim_{t\to x_0-}\frac{f(t)-f(x_0-)}{t-x_0}.$$

[My prefered notation, not yours.]

At a point of continuity this agrees with the usual definition.

A definition without any theorems is pretty useless. So perhaps you need to explore that.

E.g., What can you say about a regulated function for which $f'_{+}(x)=f'_{-}(x)=0$ at every point?

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REFERENCE: The Structure of Regulated Functions. Casper Goffman, Gadi Moran and Daniel Waterman. Proceedings of the American Mathematical Society Vol. 57, No. 1 (May, 1976), pp. 61-65.

ABSTRACT. It is shown that there is a nontrivial class of regulated functions each of which is a representable as the sum of a continuous function and a uniformly convergent series of jump functions whose jumps are those of the given function. The set of regulated functions is the union of the classes of functions of bounded ...

The regulated functions on a closed interval are those functions whose right and left limits exist at each point. Every regulated function is bounded, has a countable set of discontinuities, and is the limit of a uniformly convergent sequence of step functions. The regulated functions are of importance in the theory of stochastic processes and in the theory of everywhere convergence of Fourier series. ...