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

##### 1 Answer

This is essentially the same problem as determining the points at which any function is discontinuous, except the that points where the definition of the function changes are obvious candidates.

#### Explanation:

A function

Consider the function (one of my favourites) defined as follows:

This is continuous at every irrational number and discontinuous at every rational number.

Next define the piecewise function

#f(x) = { (0, x <= 0), (g(x), x > 0) :}#

The problem of finding where

When

When

At

#lim_(x->a-) f(x) = 0#

We also find:

#lim_(x->a+) f(x) = lim_(x->a+) g(x) = 0#

since if

Since