Let $f$ be a function. Let $a, L \in \mathbb R$. Assume that $f$ is defined on some open interval around $a$, except maybe at $a$.
There exists $𝛿 > 0$ such that for every $ε > 0$, $$0 < |x - a| < 𝛿 \implies |f(x) - L| < ε$$
Could someone explain the meaning of this statement and how it results in the function being a horizontal line?