Example

Define that

Solution

Assume ε > 0. We find a number δ > 0 such that, if 0 < | x - a | < δ, then | (mx + n) - (ma + n) | < ε.

What we need to do first is set up a connection between | (mx + n) = (ma + n) | & | x - a |. The connection is easy:

                      | (mx + n) - (ma + n) | = | m |  | x - a |.

 To make | (mx + n) - (ma + n) | < ε, we require only make | x - a | < ε/ | m |

This recommends that we select  

Therefore, if 0 < | x - a | < δ, then | (mx + n) - (ma + n) | < ε.

 Thus