Are all endpoints discontinuous? I learned that something is a limit if the left limit and right limit exist and are equal. But then doesn't this mean that if I have a function  on $[a,b]$, that the endpoints $a$ and $b$ are discontinuous because $a$ doesn't have a left limit and $b$ doesn't have a right limit?

Given $f : A \subset \Bbb R \to \Bbb R$, we say that $f$ is continuous at $x_0 \in A $ if for every $\epsilon > 0 $, exists $\delta > 0 $ such that: for all $x \in A$ that $|x - x_0| < \delta$ implies $|f(x) - f(x_0)| < \epsilon$.
The points $x$ considered in the last implication must first of all be in the function's domain. The only possibility in the endpoints $a$ and $b$ are going to $a$ from the right, and going to $b$ from the left. It makes no sense otherwise.
In general, when you have a function $f: A \subset \Bbb R \to \Bbb R $, we only consider limits of $x$ going to limit points of $A$, also called points of accumulation.
A point $x_0 \in A$ is said a point of accumulation of $A$ if for all $\epsilon > 0$, we have $(]x_0 - \epsilon, x_0 + \epsilon [ \setminus \{x_0\}) \cap A \neq \varnothing $.
