Show that the function is discontinuous at c=2 using delta epsilon

Question

I am struggling with discontinuity and continuity.

f(x)= {-1, x<2
        0, x=2 
        1, x>2

I realize that I must show that $\exists$ $\epsilon$ > 0 s.t. $\forall \delta >0$, $\exists x$, |x-2|<$\delta$ and |f(x)-0|>$\epsilon$.

I just do not know where to go from there!

I realize also you choose an epsilon, but am lost as to what to do with the delta.

Any suggestions would be great, but preferably the actual proof. I am lost on this problem and do not have any examples that have helped so far.

Thank you!

Answer 1

I realize that I must show that $\exists$ $\epsilon$ > 0 s.t. $\forall \delta >0$, $\exists x$, |x-2|<$\delta$ and |f(x)-0|>$\epsilon$.

Right. To prove this is true, we need to show there exists such an epsilon, so it suffices to prove it for $\epsilon = \frac12$. Then fix $\delta > 0$, we need to show that there is some $x$ such that $|x - 2| < \delta$ and $|f(x)| > \epsilon$. To show this is true, we can just need to show it is true for some $x$, so choose $x = 2 + \frac{\delta}{2}$...