In optimal control and differential equations we often deal with piecewise continuous functions. Here I want to extend the piecewise continuous function from an interval in $\mathbb{R}$ to a compact set in $\mathbb{R}^{n}$ for which I think there should be a standard definition. But I searched online for a long time, yet still could not find one.
Recall that a one-variable piecewise continuous function is defined as:
A real-valued function $u(t)$, $t_{0}\le t\le t_{f}$ , is said to be piecewise continuous, denoted $u\in\hat{C}[t_{0},t_{f}]$, if there is a finite (irreducible) partition $t_{0}=\theta_{0}<\theta_{1}<\cdot\cdot\cdot<\theta_{N}<\theta_{N+1}=t_{f}$ such that $u$ may be regarded as a continuous function in $[\theta_{k},\theta_{k+1}]$ for each $k=0,1,...,N$.
So my question is if I can define a multi-variable piecewise continuous function on a compact set $E\subset\mathbb{R}^{n}$ as follows:
A real-valued function $u(x)$, $x\in E$ is said to be piecewise continuous if there is a finite partition of $E$ on each coordinate such that $u$ may be regarded as a continuous function on each polytope.
The source of the question is from the book of Chachuat: Nonlinear and Dynamic Optimization, in theorem A.55 on page xv of appendix A.5: "Suppose that $f(t,x,p)$ is piecewise continuous in $(t,x,p)$...", so $f(t,x,p)$ is multivariate piecewise continuous function but Chachuat does not give the explicit definition.