I didn’t quite understand what continuity means, first i thought that the definition of continuity is the same thing as the domain of definition of the function, and what made me think like that is the definition of continuity in calculus $1$, it says :
Let $f$ be a function, $f$ Is continuous at $a$ iff $$\lim_{x\to a}f(x)=f(a)$$ So by this definition we can say that $f$ is continuous in any point in the its domain .
But when i saw some YouTube videos on that topic some of them say that the rigorous definition (using $\epsilon - \delta$ Definition) is :
Let $f:A\to \mathbb R$, where $A\subseteq \mathbb R$, $f$ Is continuous at $a\in A$ Iff $\forall \epsilon >0 , \exists \delta >0$ Such that if $$|x-a|<\delta \implies |f(x)-f(a)|<\epsilon$$ I don’t have a problem with the $\epsilon - \delta$ part of the definition, my problem is that the definition assumes that $a\in A$ but how could a function not be continuous in a point belongs to its domain of definition ?
If $f\colon\Bbb R\longrightarrow\Bbb R$ is defined by$$f(x)=\begin{cases}1&\text{ if }x=0\\0&\text{ otherwise,}\end{cases}$$then the domain of $f$ is $\Bbb R$, but $\lim_{x\to0}f(x)=0\ne f(0)$. So, no, it is not true a function $f$ is continuous at every point of its domain. Actually, if you defined $f$ by$$f(x)=\begin{cases}1&\text{ if }x\in\Bbb Q\\0&\text{ otherwise,}\end{cases}$$then $f$ is discontinuous at every point of its domain.