Sequences and series, convergence, conditional and absolute; open, closed and compact sets, Heine-Borel theorem; Abel's inequality, Dirichlet's test, Cauchy's root test, D'Alembert's ratio test, other comparisons, deMorgan's theorem applied to hypergeometric series; double series, Pringsheim's theorem.
Continuous functions, intermediate value theorem, uniform convergence (for functions), preservation of continuity under uniform convergence, Weierstrass M-test, application to power series.
Abel's theorem on continuity up to the circle of convergence, Abel's theorem on products of series, sums and products of power series, composition, inversion, differentiation and integration of functions given by power series, the principle of analytic continuation, exponential, logarithmic and trigonometric functions, periodicity of the exponential function.
Infinite products, Euler product for the Riemann zeta function, Euler's theorem on the divergence of the series of reciprocals of prime numbers.
Gamma functions, characterisation in terms of functional equation and convexity of its logarithm (see Rudin's book).
Functions on a closed interval-maximum and minimum of a continuous function, Riemann integral of a bounded function, first and second (Bonnet's) mean value theorems for integrals, improper integrals of the first and second type, absolute convergence, comparison tests, Chartier's test.
Integration of monotone functions, functions of bounded variation, their characterization as difference of increasing functions, thier integrability.
Lebesque's theorem characterizing Riemann integrable functions.
Double integrals and repeated integrals.
Example of a continuous function which is nowhere differentiable.
Orthonormal systems, trigonometric series, Fourier coefficients, Bessel's inequality, the Dirichlet kernel, partial sums of Fourier series in terms of the Dirichlet kernel, localization theorem, Fejer's theorem, Parseval's theorem, approximation of Riemann integrable functions by continuous functions (in the L^2 norm).
Weierstrass approximation theorem, algebras of functions, Stone-Weierstrass theorems for real and complex-valued functions.