Thomas' Calculus 14th Edition gives the following definition of a continuous function:
We define a continuous function to be one that is continuous at every point in its domain.
As an example, it declares the function f(x)=1/x a continuous function:
The function ƒ(x) = 1/x (Figure 2.41) is a continuous function because it is continuous at every point of its domain. The point x = 0 is not in the domain of the function ƒ, so ƒ is not continuous on any interval containing x = 0. Moreover, there is no way to extend ƒ to a new function that is defined and continuous at x = 0. The function ƒ does not have a removable discontinuity at x = 0.
On the other hand, this document I found on the MIT Math portal, has this to say about that same function 1/x:
The function 1/x is continuous on (0, ∞) and on (−∞, 0), i.e., for x > 0 and for x < 0, in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely x = 0, and it has an infinite discontinuity there.
Unless I'm misreading something here, these two sources are in direct contradiction with each other. So my questions are: is f(x)=1/x a continuous or discontinuous function, and what is the generally accepted formal definition of an overall continuous function?