Show is not uniformly continuous on .
Basically, it can't be uniformly continuous in that interval because the slope goes to (negative) infinity as x goes to zero. To prove this rigorously, remember the definition of uniform continuity: for any , there is a fixed such that implies .
To show that it's not uniformly continuous, choose any epsilon, and show that for any fixed delta, goes to infinity as x goes to 0.
There is a theorem concerning uniform continuity that says:
f is uniform continuous on its domain ,and in our case (0, ) iff for any pair of sequences on (0, { },{ } we have:
Hence to prove that : is not uniform continuous on (0, ) we must find a pair of sequences that:
,and
So if we put :
and
,then
and