real analysis - Confusion in definition of limit and continuity

I was reading from this website,

But as far as I know is not it the same as definition of continuity. For example Kenneth Ross's book has the same definition for continuity

Let $f$ be a real valued function whose domain is a subset of $\mathbb{R}$. Then $f$ is continuous at $x_0\in dom(f)$ iff for each $\epsilon>0$ there exist $\delta>0$ such that $x\in dom(f)$ and $|x-x_0|<\delta$ imply $|f(x)-f(x_0)|<\epsilon$

I am confused.

https://www.math24.net/definition-limit-function/

Answer

The only difference there is that for the existence of the limit $f(a)$ does not need to be defined.

For continuity $f(a)$ must be defined (and must be equal to the limit).