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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.