Definition:Piecewise Continuous Function with One-Sided Limits/Mistake

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A function $\psi \left({x}\right)$ is said to be piecewise-continuous in a finite interval $\left({a, b}\right)$ if:
$\text{(i)}$ the interval $\left({a, b}\right)$ can be subdivided into a finite number, $m$ say, of intervals $\left({a, a_1}\right), \left({a_1, a_2}\right), \ldots, \left({a_r, a_{r + 1} }\right), \ldots, \left({a_{m - 1}, b}\right)$, in each of which $f \left({x}\right)$ is continuous;
$\text{(ii)}$ $f \left({x}\right)$ is finite at the end-points of such an interval.

The section continues to confuse the reader by switching between $\psi \left({x}\right)$ and $f \left({x}\right)$ throughout.

It is not apparent whether there is a reason sometimes to use $\psi \left({x}\right)$ and elsewhere to use $f \left({x}\right)$, and it can only be assumed that this is an oversight, as a result of a decision to change the notation for the definition of a general piecewise-continuous function from $f \left({x}\right)$ to $\psi \left({x}\right)$ and not completely following through.