2. Continuity
Definition 0. continuous
A function is continuous at the point , if for any neighborhood of its value at a there is a neighborhood of a whose image under the mapping is contained in .
Definition 1. uniformly continuous
A function is uniformly continuous on a set if for every there exists such that for all points such that.
Mean-value theorem
Heine–Cantor theorem
if is a continuous function between two metric spaces, and is compact, then 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 's a necessary and sufficient condition for $$\forall
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