Take a real $c$, a real $\delta > 0$, and a function $f: \mathbb{R} \to \mathbb{R}$. Then there are two definitions for $f$ to have a limit at $c$.
Definition 1: If there is a real $l$ such that $$F(x) := f(x)$$ for all $0 < |x-c| < \delta$ and $$F(x) := l$$ for $x = c$ is continuous at $c$, then the function $f$ has $l$ as the limit at $c$.
Definition 2: The normal epsilon-delta definition.
My question is: how to see they are equivalent?
I think there is a mistake in Def1. But I can say that the function $F$ defined by \begin{equation} F(x)=\left\{ \begin{array}{rl} f(x), &x\neq c\\ l,&x=c \end{array} \right. \end{equation} is continuous at $c$ iff the limit of $f$ at $c$ is $l$