Analysis over R
Topology of the Euclidean Space: ball, open set, interior, adherent point, closed set, closure, density, isolated point, accumulation point, compact set, Cantor's Theorem (compact fit), Cauchy sequence, T. de Bolzano- Weierstrass, related set. Rn functions in Rm: limit, continuity, partial derivatives, directional derivatives, differentiable functions, Mean-value theorem, Leibniz's rule, Schwarz's theorem, critical points, second order optimality criteria, Lagrange multipliers, Implicit function theorem .
Metric spaces: metric topology, convergence, density, separability, isometries. Complete metric spaces. Compactness, sequential compactness. Continuous applications between metric spaces, fixed-point theorems.
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