[GolDDranks]

I would say the continuity isn't just defined at that point... I very much agree that one can't affirmatively say that the function is continuous at that point. Btw. I guess you meant to say f(x) = (x - 1)/(x - 1).

However, let's ignore that; the problem disappears if we define f(x ) = 1 at x = 1, and f(x) = 0 elsewhere.

In standard analysis that function would be discontinuous at that point.

I'm not sure about implications in the OP's kind of analysis. Would the existence of m(1) imply that there exist some smallest input that is larger than 1, that would make difference in output? Same for the largest input that is smaller than 1.