The exercise is as follows: Given a function $f : D \mapsto \mathbb R$ Are the following conditions stronger, weaker or not comparable with continuity?
$$\forall a \in D, \exists \epsilon \gt 0, \forall \delta \gt 0, \forall x \in D: |x-a| \lt \delta \Longrightarrow |f(x) - f(a)|\lt \epsilon $$
The only thing that is different from the $\epsilon$-$\delta$-definition is that the quantifiers are contrary for the $\epsilon$ and $\delta$.
Does it suffice to say that the statement is weaker than continuity because the $\exists$-quantifier is weaker than the $\forall$?
I can't come up with better arguments for it.
That was what we were taught in class but it's so self-explanatory that it shouldn't even be in our exercise lessons.
Another statement is the following:
$$ \forall \epsilon \gt 0, \exists \delta \gt 0,\forall a \in D, \forall x \in D: |x-a| \lt \delta \Longrightarrow |f(x) - f(a)|\lt \epsilon $$
It is a stronger statement and the argument that was used here is :
$Q(a,\delta) \Rightarrow \forall a \qquad \exists \delta \Rightarrow Q(\mu, \delta)$
I don't remember what the $\mu$ stands for? I think the Q is just a symbol to say it's a statement.
Please clarify this.