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

##### 1 Answer

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

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

#1/0 = oo#

#1/oo = 0#

With these kind of definitions, any rational function (apart from a few indeterminate cases e.g.