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

##### 1 Answer

#### 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