What is the difference between the terms smooth, analytical and continuous?

I saw the following (“roughly speaking”, like the author says) definition of a Lie group in ‘Group theory in Physics’, by Wu-Ki Tung:

“Roughly speaking, a Lie group is an infinite group whose elements can be parametrized smoothly and analytically.” After this, I was asking myself if I really already know the difference between these terms.

Because, what I know about 1 - analytic: if we say that a function is analytic at a point it means that its derivative is defined at this point and at the points of its neighborhood; 2 – continuous: a function is continuous at a point if you can write a neighborhood of this point where this function is still defined (and this is why those three conditions we learn in Calculus, including that one with the limit); 3 – smooth: I am not sure, but I think it is related to the differentiability of the function.

I think this may be a silly question, but I would thank you for answering

A smooth function is a continuous function with a continuous derivative. Some texts use the term smooth for a continuous function that is infinitely many times differentiables (all the $n$-th derivatives are thus continuous, since differentiability implies continuity).

An analytic function is a function that is smooth (in the sense that it is continuous and infinitely times differentiable), and the Taylor series around a point converges to the original function in the neighbourhood of that point. The existence of all derivatives doesn't imply that the Taylor series converges. A famous example is the function $$f(x)=\exp\left(\frac{-1}{x^2}\right) \text{ if } x \neq 0$$ $$f(0)=0$$
This function is continuous and infinitely many times differentiable in $x=0$. The Taylor series around this point is the constant function $T(x)=0$, so the Taylor series doesn't converge to the function $f(x)$ in the neighnourhood of $0$.