Math and science::Analysis::Tao::09. Continuous functions on R

# 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.