Prove a monotonously decreasing function with $f(p)-f(q)< \epsilon$ is continuous in a point

Question

Given are a monotonously decreasing function $f: \Bbb R \to \Bbb R$ and a point $x\in \Bbb R$. Also given is that for each $\epsilon>0$ there exist numbers $p$ and $q$, such that$f(p)-f(q)< \epsilon$. Prove that the function $f$ is continuous in $x$.

I don't even know where to start, or which theorems I could use on this one... Hints or answers are much appreciated!

Answer 1

Hint. I guess that $p<x<q$ (otherwise there are counterexamples). Then show that if $p<y<q$ then $$|f(x)-f(y)|<f(p)-f(q)< \epsilon.$$