Epsilon delta continuity I know that there are a lot of similar questions like this on this forum but still I can't figure it out one thing of this definition.
Definition: f is continuous at $x_o\in X\subset\mathbb{R}$ if
$$\forall \epsilon>0 \exists \delta>0 |\forall x| |x-x_o|<\delta\Rightarrow|f(x)-f(x_o)|<\epsilon$$
My question is, why in this definition I can't take $\epsilon \to 0$?

Please try to remember that there is no real number at all that tends to zero. You can't write, in standard analysis, anything like "consider a number $x \to 0$." 
The very definition of limit actually gives the piece of notation "$f(x) \to L$ as $x \to x_0$ " a meaning by using quantifiers: for every $\epsilon>0$ there exists $\delta>0$ etc. So, to summarize: you can arbitrarily pick a positive number, but you can't let real numbers move towards a limit value.
