## 6.2. Continuous Functions

### Example 6.2.8(b):

This function (as you could guess from its graph) is uniformly
continuous on the closed interval *[0, 1]*. To prove it, note that

| f(t) - f(s) | = | t - s | | t + s | < | t - s | 2

because *s* and *t* are in the interval *[0, 1]*. Hence,
given any * > 0* we can simply choose
* = / 10*
(or something similar) to prove uniform convergence. Can you fill in the details ? A similar
argument, incidentally, would work on the interval *[0, N]* for any number
*N*, but it would fail for the interval
*[0, )*. So, if this function then
uniformly continuous on the interval
*[0, )* ? That's the next example.