Composition and Cartesian product of continuous mappings are continuous. ... A segment of the real line is connected. An arcwise connected set is connected. ... A closed subset of a compact set is compact. ... A continulus mapping of a compact is uniformly continuous. ... Newton--Leibnitz formula. ... Taylor's formula. Taylor series expansion of some elementary functions including $(1+x)^\alpha$ for arbitraty $\alpha \in R$. ... A formula for the convergence radius of a power series. ...