Give an $\epsilon$-$\delta$ proof of the following: For all $a>0$, $f(x) = 1/x$ is continuous at $a$

Question

I'm fairly new to epsilon-delta proofs as we've just started learning them. I'm very confused about choosing a delta for this, do we have to choose an arbitrary delta with a range? Why so? Thank you very much!

Answer 1

Consider $|x-a|<a/2; a>0;$

Then $a/2 <x <(3/2)a;$

Let $\epsilon >0$ be given.

Choose $\delta =\min (a/2, \epsilon(a^2/2))$;

Then $|x-a|<\delta$ implies

$|1/x-1/a|=$

$\dfrac{|x-a|}{|xa|} \le (2/a^2)|x-a| <$

$(2/a^2)\delta < \epsilon$.