Is the following function considered piecewise-continuous??
I'm reading conflcting definitions in different places: some highlight that that the function need not be defined at the (jump/removable) discontinuities, others explicitly state that the end-points of the pieces must be well-defined. Hmm!
I think you may be confusing a "piecewise-continuous function" with a continuous "piecewise function" (that is, a function defined piecewise). Piecewise functions must be defined on the endpoints of their subdomains if they are to be continuous on a larger interval, but piecewise-continuous functions need only approach finite limits as they approach the endpoints of their subdomains. Both the Wikipedia article and the lecture notes seem to be in agreement on this, though the Wikipedia article spends more time on when a piecewise-defined function is continuous than on when a function is piecewise-continuous.