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](https://math.mit.edu/%7Ejorloff/suppnotes/suppnotes01-01a/01c.pdf) 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?