Some of the books I checked avoided the issue entirely by never using the phrase "f is continuous" without qualifying it by saying either
"f is continuous at [some point]"
"f is continuous on [some set]"
Examples of such books are:
Larson & Edwards Calculus, 9th Ed (2010)
Spivak Calculus, 3rd Ed (1994)
Stewart Calculus - Early Transcendentals, 5th Ed (2002)
On the other hand, it seems Arturo Magidin wasn't wrong. Some books say it exactly the way he did. Examples of such books are:
Adams Calculus - A Complete Course (2006)
Blank & Krantz Calculus - Single Variable, 2nd Ed (2011)
Thomas & others Calculus, 11th Ed (2004)
Their version of "f is continuous" yields the advantage of being consistent with definition of continuous function used in later courses such as Topology, traded off against the disadvantage of being perhaps inconsistent, in some cases, with the earlier notion of continuous function introduced in Precalculus.
But since at least _some_ books match Arturo's version perfectly, and _none_ match the definition I proposed, I withdraw my proposed definition.
Still, all in all, I think it makes sense, at the Elementary Calculus level, to avoid potential confusion by never just saying "f is continuous", but rather always qualifying it using one of the forms
"f is continuous [at some point]"
"f is continuous [on some set]"
except in cases where omitting the qualification would not contradict earlier Precalculus notions.