Given the monotone decreasing function and a point $a\in\mathbb{R}$ prove that f is continuous in a.

Question

Given the monotone decreasing function and a point $a\in\mathbb{R}$, given is that $\forall \epsilon \exists p<a,q>a$ so that $f(p)-f(q)<\epsilon$ prove that f is continuous in a.

where do I go from here?

Answer 1

Let $\epsilon >0$. Then let $p_\epsilon$ and $q_\epsilon$ be the points guaranteed by your given condition. Let $\delta=\min\{a-p_\epsilon,q_\epsilon-a\}$. Then we have:

$|x-a|<\delta \implies p_\epsilon\leq x\leq a$ or $a\leq x\leq q_\epsilon$.

In either case, can you show that $|f(x)-f(a)|<\epsilon$?