Checking uniform continuity of a function.

Question

I need to check whether the function $f : (0,1) \to \Bbb{R}$ given by $f(x)= \frac{e^{-1/x}}{x}$ is uniformly continuous on its domain.

I know it is continuous since it is a composition of continuous functions. But I have no idea whether it is uniformly continuously or not. I am confused as to how one speculates about whether it will be or not by just looking at it.

I assumed that it is and I tried to estimate $|f(x)-f(y)|$ where $x$ and $y$ are in the domain but I can't proceed. Any help is appreciated. Thanks. :)

Answer 1

If you can extend $f$ to $[0,1]$ in such a way that the extension is continuous you are done, since continuos functions on compact sets are uniformly continuous. The extension to $x=1$ is obvious. Can you see how to extend it t0 $x=0$?