Question

How do I find the point(s) at which a given rational function is discontinuous?

Answer 1

It depends...

Explanation:

It depends on what definition of continuity/discontinuity you have been given.

Continuity can be roughly understood as being able to draw a graph of a function without lifting your pen. With that understanding, a rational function will be continous except at the points where its denominator is zero.

When the denominator is zero, a rational function either has a vertical asymptote or a hole.

While a rational function cannot be considered continuous at a value of `x` where it has a hole or vertical asymptote, note that such points are not actually part of the domain of the function.

A more formal definition of continuity only concerns points in the domain of the function.

With that approach any rational function is continuous at all points of its domain.

Advanced footnote

Given any rational function with real coefficients, we can consider it as a function from `RR uu { oo } = RR_oo` to `RR_oo` where the object `oo` satisfies some properties like:

`1/0 = oo`

`1/oo = 0`

With these kind of definitions, any rational function (apart from a few indeterminate cases e.g. `f(x) = 0/0`) is well defined and continuous on the whole of `RR_oo` (known as the real projective line).