## 6.2. Continuous Functions

### Example 6.2.8(c):

While this function**is**uniformly continuous on any interval

*[0, N]*(

*N*any number) it is no longer uniformly continuous on the interval

*[0, )*. To prove this, take

*= 1*. Note that

| f(s) - f(t) | = | s - t | | s + t |

Can you see that if *s = t + * and
if *t* is sufficiently large (depending on the undetermined
)
then no matter what is chosen,
*| f(s) - f(t) | > 1*. That would prove that the function is
no longer uniformly continuous. The details are left as an exercise.

Note that this argument no longer works on a bounded interval
*[0, N]*. Here we can not make *t* 'sufficiently large', since it can
be no larger than *N*. And indeed, the function is uniformly continuous
on those bounded intervals. Later we will show that any function
that is continuous on a compact set is necessarily uniformly continuous.