Notes on Uniform Convergence by Prof. B. V. Limaye:
Review of basic concepts of real numbers: Archimedean property, Completeness.
Metric spaces, compactness, connectedness, (with emphasis on Rn).
Continuity and uniform continuity.
Monotonic functions, Functions of bounded variation; Absolutely continuous functions. Derivatives of functions and Taylor's theorem.
Riemann integral and its properties, characterization of Riemann integrable functions. Improper integrals, Gamma functions.
Sequences and series of functions, uniform convergence and its relation to continuity, differentiation and integration. Fourier series, pointwise convergence, Fejer's theorem, Weierstrass approximation theorem.