# Pick out the correct choices for a continuous function

let $f(x)=\dfrac{e^{\large \frac{-1}{x}}}{x}$ where $x\in (0,1)$ .then on $(0,1)$

1.$f$ is uniformly continuous

2.$f$ is continuous but not uniformly continuous

3.$f$ is bounded

4.$f$ is not continuous

I found it to be continuous but not uniformly as $\lim_{x\rightarrow 0} f(x)$ does not exist.It is bounded below by 0 and above by 1.Am i right? Would be grateful if someone could suggest me required edits.

• $$\lim_{x\to 0^+}\frac{e^{-1/x}}x\stackrel{y=\frac1x}=\lim_{y\to\infty } \frac y{e^{y}}\stackrel {l'H}=\lim_{y\to\infty}\frac1{e^y}=0$$ so the limit exists (finitely) when $\;x\to 0^+\;$ – Timbuc Jan 15 '15 at 16:16
• thanks @Timbuc for your answer – Learnmore Jan 16 '15 at 2:28

$$\lim_{x\to 0}\frac{e^{-\frac{1}{x}}}{x}=0$$ therefore the function is uniformly continuous because you can prolonge $x\mapsto \frac{e^{-\frac{1}{x}}}{x}$ to $0$ and the prolongement is continuous on $[0,1]$. But you know that a continuous function on a compact is uniformly continuous, therefore it's uniformly continuous on $[0,1]$ and thus on $]0,1[$.