I would generally want the intervals to cover the entire domain (or rather for their closures to cover it). And half-infinite (or all of R) intervals would also

I would generally want the intervals to cover the entire domain (or rather for their closures to cover it). And half-infinite (or all of R) intervals would also be allowed, but I couldn't think of a good way to state that concisely. And I would only require the limits at the finite end points so that boundedness is not a concern (piecewise continuity should be a local property anyway). Surprisingly complicated to specify fully!