[Math] But what is a continuous function

Question

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.

Answer 1

Mathematicians (but not all calculus books) mean "continuous at every point of its domain" when they say a function is "continuous." The functions $f(x) = 1/x$ and $f(x)=\ln x$ are continuous functions.