Function convergence at a point, definition
ε-closeness, of a function to a real
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that the function \( f \) is ε-close to \( L \) iff [...].
Local ε-closeness, of a function to a real
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( x_0 \) be an adherent point of \( X \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that \( f \) is ε-close to \( L \) near \( x_0 \) iff [...] such that [...] when restricted to [...].
Convergence of a function at a point
Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) be a function, let \( E \) be a subset of \( X \), let \( x_0 \) be an adherent point of \( E \), let \( L \) be a real, and let \( \varepsilon > 0 \) be a real.
We say that \( f \) converges to \( L \) at \( x_0 \) in \( E \), and write \( \lim_{x \rightarrow x_0; x \in E} f(x) = L\), iff \( f \), after [...], is [...] for every \( \varepsilon > 0 \). If \( f \) does not converge to any number \( L \) at \( x_0 \), we say that \( f \) [...], and leave \( \lim_{x \rightarrow x_0; x \in E} f(x) \) undefined.
These definitions are separated into 2 paragraphs due to the many supporting objects required.