TITLE: Point of discontinuity
QUESTION [11 upvotes]: I have a function: 
$$f(x) = x$$
Defined over the domain $\mathbb{R} \backslash 0$.
Is it correct to say that:
The function is continuous, but it has a point of discontinuity at $x=0$?

REPLY [2 votes]: If $x=0$ is not part of the domain of the function, there is no sense in talking about the properties of the function at that point - continuity or anything else.
It happens that the domain on which this function is defined is a part of a larger domain - it is a fundamental issue in mathematics to identify the possibility of extending functions to such larger domains in "good" ways - either preserving useful properties like continuity, or acquiring new ones - like connectedness, compactness or roots to specific equations.
Your $f$ could be extended to $\mathbb R$ by defining $f(0)=\pi$. If you want to preserve continuity, however, you need to define $f(0)=0$.
You might consider the function $g(x)$ defined on the same domain with $g(x)=-x$ when $x$ is negative, and $g(x)=x$ when $x$ is positive. Defining $g(0) = 0$ keeps $g$ continuous, but it is no longer differentiable over the whole domain of definition. Sometimes there is a trade-off between the properties you want and the domain you choose.