> For the complete documentation index, see [llms.txt](https://coldmoon.gitbook.io/mathematical-analysis/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://coldmoon.gitbook.io/mathematical-analysis/mathematical-analysis-1/2.-continuity.md).

# 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&#x20;

$$\blacksquare$$

end


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