But what is a continuous function? I have a very basic problem. I am confused about "continuous function" term. 
What really is a continuous function? A function that is continuous for all of its domain or for all real numbers?
Let's say:
$\ln|x|$ - the graph clearly says it's continuous for all real numbers except for $0$ which is not part of the domain. So is this function continuous or not? I could say same about $\tan{x}$ or $\frac{x+1}{x}$
And also what about:
$\ln{x}$ - the graph clearly says it's continuous for all of its domain: $(0; \infty)$ - so is this $f$ continuous or not?
Thanks for clarification.

It is, but what you are looking for might be the notion of continous extension.
Here both $\frac{1}{⋅}$ and $\log$ are continuous in the sense that they are (pointwise) continuous on their respective domains, as is
$$
\left|
\begin{array}{lll}
    f : &\mathbb{R}^* ⟶ \mathbb{R}\\
        & x \longmapsto x²
\end{array}
\right.
$$
The difference is that there exists $g: \mathbb{R}→\mathbb{R}$ continuous such that $g(x)=f(x)$ for all $x∈\mathbb{R}^*$ — so $f$ has a continous extension on $\mathbb{R}$ — whereas $\frac{1}{⋅}$ has no such thing.
