Yu.Burman

Calculus, term 1

Course synopsis (principal theorems)

1. Composition and Cartesian product of continuous mappings are continuous.
2. Two definition of the limit of function (by means of continuity and by means of test sequences) are equivalent.
3. Any set bounded from above have the least upper bound.
4. A segment of the real line is connected. An arcwise connected set is connected.
5. Theorem on embedded segments.
6. Mean value theorem.
7. A closed subset of a compact set is compact.
8. A metric compact contains a finite $\eps$-net for any $\eps > 0$.
9. Heine--Borel lemma.
10. Metric spaces $R$ and $C[0,1]$ are complete.
11. Changing the order of passing to limits, if the convergence is uniform.
12. A continulus mapping of a compact is uniformly continuous.
13. A continuous function is Riemann integrable.
14. Lagrange's and Cauchy's intermediate point theorems.
15. Newton--Leibnitz formula.
16. The contraction mapping principle.
17. A local solvability of a differential equation.
18. An explicit formula for solution of a linear differential equation with constrant coefficients.
19. Taylor's formula.
20. Taylor series expansion of some elementary functions including $(1+x)^\alpha$ for arbitraty $\alpha \in R$.
21. Mixed partial derivatives are independent of the order of differentiation.
22. A formula for the convergence radius of a power series.
23. A differentiability of the series.
24. Partition of a unity.
25. Description of maximal closed ideals in an algebra of infinitely smooth functions.