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Let $E subset mathbb R$, and $f:Erightarrow mathbb R$, be a map. Set $$Delta_{varepsilon,a} = {delta>0: B(f(a),delta)supset f(Ecap B(a,varepsilon))} for varepsilon>0,ain E$$
$$A_{n,delta}={ain E:Ecap B(a,delta)⊂ f^{−1}(B(f(a),frac{1}{n}))} for n in mathbb N, delta>0$$
where $B(x,r)$ denotes the interval $(x-r,x+r)$(i.e. ball of radius r in real metric space).Then,

$(i)$If $$bigcup_{varepsilon>0}bigcap_{ain E}Delta_{varepsilon,a} = (0,infty)$$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 $etain cap_a Delta_{epsilon, a}$, this says $|x-a|<epsilon$ implies $|f(x)-f(a)|<eta$...)
ii. says $f$ is continuous on $E$.