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Pointwise Continuity and Sequential Tests
Learn pointwise continuity in metric spaces using epsilon-delta definitions, isolated points, and sequential tests.
UG6 Jul 2026
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Study continuous mappings and fixed point principles in metric spaces with definitions, theorems, examples, exercises, and answers.
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Continuous Mappings and Fixed Point Principles Learning Sequence. 22 concepts.
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Learn pointwise continuity in metric spaces using epsilon-delta definitions, isolated points, and sequential tests.
Study limits of mappings in metric spaces, sequential criteria, and the exact relation between limits and continuity.
Use open and closed set continuity tests through inverse images, neighbourhoods, interiors, and bases in metric spaces.
Learn composition of continuous mappings, closure characterizations, pasting tests, and separable image results.
Study coordinate mappings, vector-valued continuity, stereographic formulas, discrete domains, and integral operators.
Learn continuous extension from dense and closed sets using equality sets, unique extensions, and limit criteria.
Study continuous scalar functions under sums, products, scalar multiples, moduli, reciprocals, and functionals.
Learn uniform continuity, distance from a set, closure by distance, and separation of disjoint closed sets.
Study uniformly continuous mappings, preservation of Cauchy sequences, and dense-subset extension into complete spaces.
Learn homeomorphism, topological equivalence, open and closed mappings, and properties not preserved by homeomorphism.
Study isometry, isometric metric spaces, completeness preservation, sequence shifts, and isometric embeddings.
Learn equivalent metrics through sequence convergence, identity maps, Euclidean comparisons, and non-equivalent examples.
Study pointwise convergence of function sequences, spike examples, powers, and Baire restrictions on discontinuities.
Learn uniform convergence, supremum criteria, continuity of uniform limits, and the uniform Cauchy criterion.
Study uniformly convergent series of functions, partial sums, continuity of sums, and non-uniform counterexamples.
Apply the Weierstrass M-test to trigonometric, power-type, and step-function series with uniform convergence.
Learn the Tietze extension theorem using distance functions, bounded approximation, and uniform convergence.
Study contraction mapping, fixed points, the Banach contraction principle, uniqueness, and iterated contractions.
Solve linear systems by fixed point methods using contraction criteria in maximum, sum, and Euclidean metrics.
Apply Picard iteration and contraction methods to differential equations, integral equations, and uniqueness results.
Study iterated contractions, Volterra operators, exact power estimates, and fixed point consequences for iterates.
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Comprehensive module covering 5 sections in Functional Analysis.