2. Continuity

Definition 0. continuous

A function $f$ is continuous at the point $a$, if for any neighborhood $V (f (a))$ of its value $f (a)$ at a there is a neighborhood $U(a)$ of a whose image under the mapping$f$ is contained in $V (f (a))$.

Definition 1. uniformly continuous

A function $f : E \to \mathbb R$ is uniformly continuous on a set $E \subset \mathbb R$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|f (x_1)−f (x_2)| < \varepsilon$for all points $x_1,x_2\in E$ such that$|x_1 − x_2| < \delta$.

Mean-value theorem

Heine–Cantor theorem

if $f : M \to N$ is a continuous function between two metric spaces, and $M$ is compact, then $f$ is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.

Heine theorem

The limit $\lim _{x \rightarrow a} f(x)=b$'s a necessary and sufficient condition for $$\forall

$\blacksquare$

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