The exhaustive list of topics in Modern Analysis in which we provide Help with Homework Assignment and Help with Project is as follows:

- Sets: Sets of numbers, set operations, family of subsets, constructing
- new sets from old ones, De Morgan’s laws.
- Maps: Surjection, injection, invertible maps, direct and inverse images
- and their relations, inverse of a map, composition of maps, restriction
- of a map.
- Finite and infinite sets, countable and uncountable sets, Results related
- to countable sets.
- Countability of rational numbers.
- Absolute value of a real number, epsilon neibourhood of a point,
- supremum and infimum and properties. Completeness property of the
- real numbers. Archimedean property, Density of rational numbers.
- Nested intervals property, uncountability of reals..
- Sequences, convergence, examples.
- Conditions of convergence, monotone sequence, examples.
- Tests of convergence of sequences of real numbers.
- Subsequences, Bolzano-Weierstrass theorem, Cauchy’s criterion.
- Infinite series, tests of convergence.
- Definition and examples of metric spaces.
- Open ball, neighborhood of a point, open set, closed set, and their
- properties.
- Interior, exterior and their properties.
- Closure of a set and their properties.
- Limit point of a subset and limit of a sequence in a metric space,
- Boundary of a set and properties.
- Equivalence of metric spaces. Product spaces.
- Continuity of a map between two metric spaces and examples.
- Various characterizations continuous maps.
- Cauchy sequence, Complete metric spaces. Example of complete and
- incomplete metric spaces.
- Completeness of the space of real numbers.
- Compactness, examples. Continuous image of a compact space.
- Heine-Borel theorem.
- Separation. Connectedness, connected subsets of the real metric space.
- Properties of connected spaces. Examples. characterization of
- connectedness. Intermediate value theorem.
- Differentiable functions, higher derivatives, Leibnitz Rule.
- Rolle’s theorem, Mean value theorems.
- L’ Hospital rule. Examples.
- Taylor’s formula and Taylor series.
- Monotone functions, maxima and minima.
- Riemann Integral, example of functions which are not Riemann
- integrable.
- Criterion of Riemann integrability, Properties of Riemann integral.
- Fundamental theorems of integral calculus.
- Integration by parts.
- Trapezoidal rule and Simpsion’s rules of approximation of a Riemann
- integrals.
- Improper integrals.
- Sequence and Series of functions, uniform convergence of functions.
- Weierstrass M-test.
- Power Series, radius of convergence.
- Properties of functions represented by real power series.
- Fourier Series.
- More on Fourier Series.