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Yu.Burman

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Calculus–1

- Composition and cartesian product of continuous maps is continuous.
- A set is closed iff it contains all its limit points.
- The operations over reals are defined soundly.
- Completeness of
**R** and its corollaries (existence of the supremum,
etc.)
- The segment is a connected set. An arcwise connected space is connected.
- The intermediate value theorem.
- Compacts in
**R**^{n} (the Heine?ndash;Borel lemma).
- An infinite subset of a compact has a limit point.
- Exchange of limits when the convergence is uniform.
- A uniform limit of a family of continuous maps is continuous.
- Existence and uniqueness of the metric completion.
- A continuous map on a compact set is uniformly continuous.
- A continuous function is Riemann integrable.
- Lagrange's and Cauchy's theorems on intermediate points.
- The Newton–Leibnitz formula.
- The Taylor's formula.
- Convergence of the Taylor series for princial elementary functions.
- Mixed derivative do not depend on the order of differentiation.
- The completeness of $C(X)$.
- The contraction mapping principle.
- Local solvability of a differential equation.
- The partition of unity.
- Maximal closed ideals in the algebras of continuos and smooth functions.
- All the norms in a finite-dimensional space are equivalent.
- The inverse and implicit function theorems.