I feel like I do understand $f(x)$'s continuity at the point $c$ : $lim_{x \to c}f(x) = f(c)$. Though general idea seems to be clear, I still have some kind of doubts whether it is sufficient. The fact that $x \to c$ does not specify how exactly it approaches $c$. It does not require $x$ to take all possible values $(x_0, c)$ explicitly (but theoretically you should check each and every value from given interval).
So my questions is: am I wrong somewhy? Do I misunderstand the fact that definition above is indeed sufficient? Should not $\lbrace x \to c \rbrace$ take all possible values? Or I am right and that particular definition is kind of "common agreement" on what it requires implicitly?..
For the continuity at a point $c$ it doesn't matter how $x$ approches to $c\in dom(f)$.
The key idea of the definition is that whatever $\epsilon>0$ you choose then you can always find a $\delta>0$ such that:
$$\forall x\in dom(f) \quad |x-c|<\delta \implies |f(x)-f(c)|<\epsilon$$
whch is equivalent to the condition
$$\lim_{x \to c} f(x) = f(c)$$