explain why every uniformly continuous function is not bound

explain why every uniformly continuous function is not bounded.

Solution

Defintion of Uniform Continuous Function:

A function f:IR is said to be Uniformly Continuous on I if >0 >0 such that if x,yI satisfy xy< then

f(x)f(y)<.

And it can be bounded, reason is given below.

The function f(x)= x is unbounded on R, but uniformly continuous on R. The function f(x)=sqrt(x) is another example.

If I is a bounded interval and f:IR is uniformly continuous, then is f bounded . Find >0 such that for |xy|<|, |f(x)f(y)|<1. Then by partitioning the interval I up into a finite number of pieces smaller than , you can show f is bounded.

The same holds true if I is any bounded set, not just an interval.