Let $E \subset \mathbb R$, and $f:E\rightarrow \mathbb R$, be a map. Set $$\Delta_{\varepsilon,a} = \{\delta>0: B(f(a),\delta)\supset f(E\cap B(a,\varepsilon))\}\ for\ \varepsilon>0,a\in E$$ $$A_{n,\delta}=\{a\in E:E\cap B(a,\delta)⊂ f^{−1}(B(f(a),\frac{1}{n}))\}\ for\ n \in \mathbb N, \delta>0$$ where $B(x,r)$ denotes the interval $(x-r,x+r)$(i.e. ball of radius r in real metric space).Then,
$(i)$If $$\bigcup_{\varepsilon>0}\bigcap_{a\in E}\Delta_{\varepsilon,a} = (0,\infty)$$Then what is the best that you can say about f without any extra assumption on the hypothesis?
$(ii)$If $$\bigcap_{n \in \mathbb N}\bigcup_{\delta>0}A_{n,\delta} = E$$ Then what is the best that you can say about f without any extra assumption on the hypothesis?
I am not getting the notation in this question about lots of unions and intersections and also I have a hunch that both of these imply continuity or uniform continuity or something similar. Please help. Thanks in advance.
Yes, i. says $f$ is uniformly continuous on $E$, (for example, for any $\eta>0$, there is $\epsilon$ so that $\eta\in \cap_a \Delta_{\epsilon, a}$, this says $|x-a|<\epsilon$ implies $|f(x)-f(a)|<\eta$...) ii. says $f$ is continuous on $E$.