Analysis: Composition of a regulated function and a continuous function

Question

What would be an example which proves that the composition of a regulated function $f: [-1, 1] \to \mathbb{R}$ and a continuous function $g: [-1, 1] \to [-1, 1]$, of the form $f \circ g$ may not necessarily be regulated? And how is it so?

Answer 1

Here is the example from http://users.math.cas.cz/~tvrdy/Cichon.pdf

"In contrast to the case of continuous functions the composition of regulated functions need not to be regulated. The simplest example is a composition $(g \circ f)$ of functions $f, g: [0,1] \to \mathbb R: f(x) = x·sin(1/x)$ and $g(x) = sgn( x) $ ($f$ continuous and both regulated), which has no one-side limits at 0. Thus even a composition of a regulated and continuous functions need not to be regulated".
(the sign function has values $+1$ for $x > 1$; $0$ for $x = 0$ and $-1$ for $x < 1$)

How it is so ? is interesting.......

Suppose that $g$ is regulated and has left and right limits $g(a-)$ and $g(a+)$ at a point $a$, but the limits differ, and $f$ is continuous with some $x$ having $f(x) = a$. While $f$ is continuous at $x$, $f(x)$ can oscillate either side of $a$ as x is approached from the right (getting ever closer). But then $g(f(x))$ will oscillate between values ever closer to the left and right limits $g(a-)$ and $g(a+)$ which differ. So, $g(f(x))$ with not have a right limit as x is approached from the right.
This is what is happening in the example.

For interest, the opposite composition with a continuous function of a regulated function is regulated. See for example Composition of regulated functions on $[0,1]$}