Università Roma Tre

Open, closed and compact sets. Weierstrass Theorem. Uniformly continuous functions.
Differentiability of functions. Rules for computing derivatives. Derivatives and monotonicity. Fundamental theorems on derivatives (Fermat, Rolle, Cauchy, Lagrange). Theorem of Bernoulli-Hopital. Critical points. Second derivative. Convex functions. Qualitative study of functions. Successive derivatives and Taylor's formula. Use of Taylor's formula in computing limits.
Riemann's integral: partial sums, integrability. Integrability of monotone and piecewise continuous functions. Computation of primitives. Fundamental theorem of calculus. Integral remainder in Taylor's formula. Improper integrals; comparison with series.
Complex numbers, exponential series in the complex plane and fundamental theorem of algebra.

Luigi Chierchia, Corso di Analisi, prima parte, Una introduzione rigorosa all'analisi matematica su R.