Continuity of 1/x

Question

I am confused with what $8(ii)$ wants from me, I answered the first part of this question with help from the question posted here
Is $f(x)=1/x$ continuous on $(0,\infty)$?
But the this proves continuity and works for all $\epsilon>0$, so how do I prove it doesn't for $\epsilon=1$??

Answer 1

The general proof for all $\epsilon>0$ shows that specifically for $\epsilon=1$: $$\forall c\in(0,\infty)\colon \exists\delta>0\colon \forall x\in(0,\infty)\colon |x-c|<\delta\Rightarrow |f(x)-f(c)|<1 .$$ Your task is to show that the slightly rearranged claim $$\exists\delta>0\colon\forall c\in(0,\infty)\colon \forall x\in(0,\infty)\colon |x-c|<\delta\Rightarrow |f(x)-f(c)|<1 $$ is false.

Answer 2

The order in which you take $\delta$ and $c$ is the key! Read carefully the last sentence of (i).