## Yu.Burman

## Calculus, term 1

### Course synopsis (principal theorems)

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