## 6.2. Continuous Functions

### Example 6.2.4(b):

*{x*is any sequence of numbers (rational or irrational) that converges to zero, then there exists an integer

_{n}}*N*such that

*|x*for

_{n}| <*n > N*. But

*f(x*is either zero or

_{n})*x*itself, and in any case we have

_{n}That proves that the sequence of| f(x_{n}) | | x_{n}| <

*{x*converges to

_{n})}*0 = f(0)*, which proves that the function is continuous at zero.

As an exercise, prove that the function is not continuous for
any other *x*.