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.

The exact answer depends on your chosen definition of "function" (there is more than one). For most uses, a function is regarded as being continuous on an interval $(a,b)$ if for every number $c$ in $(a,b)$, $f(x)=\lim_{x\to c} f(x)$.
In your example $f(x)=\ln{x}$ is continuous on the interval $(0,\infty)$ and either undefined or complex/multivalued everywhere else, depending on whether you consider the codomain (range) of $f$ to include the complex numbers or not.
In other words, no function is ever just 'continuous' - it is continuous within an interval (which may or may not be its domain).
