Website owner:  James Miller

Theory of Functions of a Real Variable

Set theory. Union, intersection, complement, difference. Venn diagram. Algebra of sets. Countable set. Cardinality. Product set. Partition.

Number systems. Rational, irrational and real numbers. Open and closed intervals. Fields. Least upper bound. Greatest lower bound. Axioms, laws and properties.

Point sets in one, two, three and n-dimensional Euclidean spaces. Intervals, neighborhoods, closed sets, open sets, limit points, isolated points. Interior, exterior and boundary points. Derived set. Closure of a set. Perfect set. Arcwise connected sets. Regions. Coverings. Theorems. Bounded, compact sets.

Functions

Continuous functions. Sequences. Accumulation point. Limit superior and inferior. Cauchy sequence. Monotonic sequences. Nested intervals. Cantor's principle. Metric space. Uniform convergence of sequences of functions. Theorems.

Measure theory. Measure of a point set. Open covering. Exterior and interior measure. Theorems. Borel sets.

Measurable functions. Theorems. Baire classes. Egorov's theorem.

The Lebesgue integral. Theorems. Bounded, dominated, monotone convergence theorems.

Differentiation and integration. Monotonic functions. Jumps at discontinuities. Functions of bounded variation. Absolutely continuous functions. Theorems.

Lp spaces, Hilbert space. Schwartz's, Holder's, Minkowski's inequalities. Convergence in the mean. Cauchy sequences. Riesz-Fischer theorem. Convergence in measure.