f(x) = 1/x: to be or not to be continuous ...

Greetings!

Nowadays, most calculus books say that f(x) = 1/x is discontinuous in
x = 0. However, "analytic" oriented books (like Apostol) say f(x) = 1/
x is continuous: the point 0 doesn't matter, since it doesn't belong
to the function's domain.

I'm really curious to know when and who made this bifurcation. What
concept came first? Any historical references?

Thanks in advance, Humberto.

.

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