Yu.Burman

Calculus–1

1. Composition and cartesian product of continuous maps is continuous.
2. A set is closed iff it contains all its limit points.
3. The operations over reals are defined soundly.
4. Completeness of R and its corollaries (existence of the supremum, etc.)
5. The segment is a connected set. An arcwise connected space is connected.
6. The intermediate value theorem.
7. Compacts in Rⁿ (the Heine?ndash;Borel lemma).
8. An infinite subset of a compact has a limit point.
9. Exchange of limits when the convergence is uniform.
10. A uniform limit of a family of continuous maps is continuous.
11. Existence and uniqueness of the metric completion.
12. A continuous map on a compact set is uniformly continuous.
13. A continuous function is Riemann integrable.
14. Lagrange's and Cauchy's theorems on intermediate points.
15. The Newton–Leibnitz formula.
16. The Taylor's formula.
17. Convergence of the Taylor series for princial elementary functions.
18. Mixed derivative do not depend on the order of differentiation.
19. The completeness of $C(X)$.
20. The contraction mapping principle.
21. Local solvability of a differential equation.
22. The partition of unity.
23. Maximal closed ideals in the algebras of continuos and smooth functions.
24. All the norms in a finite-dimensional space are equivalent.
25. The inverse and implicit function theorems.