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Mathematical and Real Analysis
Explore real numbers and limits; continuity and compactness; functions and integration. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Quantifiers
Universal and existential quantifiers specify how a statement ranges over a domain.
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Real numbers and limits
Completeness of the real numbers
Every nonempty real set bounded above has a least upper bound.
Video explanations to be addedSequence convergence
A sequence converges when its terms eventually lie within every prescribed distance of a limit.
Video explanations to be addedCauchy sequences
Terms of a Cauchy sequence eventually become arbitrarily close to each other.
Video explanations to be addedLimit superior and inferior
Tail suprema and infima describe the extreme limiting behavior of a sequence.
Video explanations to be addedBolzano–Weierstrass theorem
Every bounded sequence in finite-dimensional Euclidean space has a convergent subsequence.
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Continuity and compactness
Continuity
Continuity at a point means that the function value equals its limit there; continuity on an interval requires this at each relevant point.
3 reviewed resourcesUniform continuity
A single input tolerance controls output error uniformly across the domain.
Video explanations to be addedEpsilon–delta limits
Every prescribed output tolerance must follow from a sufficiently small nonzero input displacement.
Video explanations to be addedIntermediate value theorem
A continuous real function on an interval takes every value between two of its values.
Video explanations to be addedExtreme value theorem
A continuous real function on a nonempty compact set attains its maximum and minimum.
Video explanations to be addedMetric spaces
A metric assigns nonnegative distances satisfying separation, symmetry and the triangle inequality.
Video explanations to be addedCompactness
A space is compact when every open cover admits a finite subcover.
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Functions and integration
Pointwise convergence
A sequence of functions converges separately at each point, with tolerances allowed to depend on the point.
Video explanations to be addedUniform convergence
One index bounds the approximation error at every point of the domain.
Video explanations to be addedRiemann integrability
Riemann integrability requires upper and lower sums to approach the same value for bounded functions.
Video explanations to be addedPower series
A power series converges inside its convergence radius; boundary points require separate tests.
Video explanations to be addedInterchanging limits
Exchanging limits, derivatives or integrals requires hypotheses such as suitable uniform convergence.
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