Continuity on $\mathbb R$

Question

I’ve seen 3 different definitions for continuity from different lecturers? Could someone illuminate me about which one is true. Thanks in advance

Let $I$ is domain of $f$

$f$ is continuous on the point $a$ iff?

1) $\forall \varepsilon>0$ $\forall x \in I$ $\exists \delta _{x,\varepsilon} >0$ $|x-a| < \delta _{x,\varepsilon} $ $\Rightarrow$ $|f(x)-f(a)|< \varepsilon$

2) $\forall \varepsilon>0$ $\exists \delta _{\varepsilon} >0$ $\forall x \in I$ $|x-a| < \delta _{\varepsilon} $ $\Rightarrow$ $|f(x)-f(a)|< \varepsilon$

3) $\forall \varepsilon>0$ $\exists \delta _{a,\varepsilon} >0$ $\forall x \in I$ $|x-a| < \delta _{a,\varepsilon} $ $\Rightarrow$ $|f(x)-f(a)|< \varepsilon$

Answer 1

Maja Blumentein correctly comments:

The only difference between 2 and 3 is subscipt $δ_{a,\varepsilon}$. Number $δ$ depends on both $a$ and $\varepsilon$ so apparently one lecturer wanted to explicitly denote that, while the other one thought it wasn't necessary. Therefore it's clear these two are equivalent and the only difference is notation.

On the other hand (1) does not work at all. It declares everything to be continuous, because once $x$ is given you can always chose $\delta_{x,\varepsilon}$ to be larger than $|x-a|$, which automatically makes the $\Rightarrow$ true no matter what the function is doing.

Definitions (2) and (3) are correct.