Show that $f(x)=\ln(x)$ is not uniformly continuous on $(0,\infty)$ I'm trying to show that $f(x)=\ln(x)$ is not uniformly continuous on the interval $(0,\infty).$
This is what I have so far:
Let $\epsilon=1.$
Choose $\delta=$
if $x,y\in(0,\infty)$ with $|y-x|<\delta$ then $|f(y)-f(x)|=|\ln\left(\frac{y}{x}\right)|$
I'm stuck at this point though, are there any well known inequalities I can use here?

General fact: No unbounded function on a bounded subset of $\mathbb R$ can be uniformly continuous. (In the problem at hand, we have $\ln x $ unbounded on $(0,1),$ so it can't be uniformly continuous there.)
