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

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

The section continues to confuse the reader by switching between $\map \psi x$ and $\map f x$ throughout.

It is not apparent whether there is a reason sometimes to use $\map \psi x$ and elsewhere to use $\map f x$, 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 $\map f x$ to $\map \psi x$ and not completely following through.