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# Theory of Functions of a Real Variable

Select from the following:
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.

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