||Compactness and several characterizations in metric spaces. The Baire Category Theorem. Continuity, including connections with connectedness, compactness, and uniform continuity. Discontinuities of real functions; monotonic functions. Differentiability and rigorous treatment of the main theorems of differential calculus, including Darboux's Theorem, L'Hospital's Rule, and Taylors' Theorem. The Darboux formulation of the Riemann Integral; Riemann's formulation and their equivalence; the Lebesgue characterization of Riemann integrability. The Fundamental Theorem of Calculus and integration by parts. Improper integrals and the Cauchy and Abel-Dirichlet criteria. Sequences and series of functions; pointwise and uniform convergence with applications to continuity, integration, and differentiation; Dini's Theorem. Power series and radius of convergence. Spaces of functions; the Stone-Weierstrass Theorem; the Arzela-Ascoli Theorem. The Banach Contraction Mapping Fixed Point Principle with application to ODEs. Briefly: Inner products on periodic functions, L^2 convergence of Fourier series and Plancherel's Theorem.