Is $f(x)=\frac{1}{x}$ uniformly continuous then x $\in (0,1)$?
I think that it's not uniformly continuous so I am trying to prove that there exists an epsilon>0 for all deltas>0 and there exist x,y such that
$|f(x)−f(y)|≥ϵ$ if $ |x−y|<δ$
I started by choosin epsilon=1 . Then:
$|\frac{1}{x}−\frac{1}{y}|=\frac{|x−y|}{xy}$
Now, I think, I need to choose such values of x and y expressed through delta such that the equality above would be greater or equal than one, but I am having trouble thinking of such values. Is my approach any good?