Number sets: natural numbers and induction, summation symbol and sum of the first n integers. Factorials and binomial coefficients. Binomial formula. Rational numbers, irrationality of square root of 2. Ordering and sup property, axiomatic property of real numbers. Decimal expansions.
Complex numbers. Algebraic representation and geometric interpretation. Conjugate and absolute value. Trigonometric and exponential representations. N-th roots. Equations in C. Fundamental Theorems of Algebra.
Introduction to the topology of the real line. Real valued sequences and epsilon-delta definition of their limit. Converging, diverging and irregular. Properties of limits: uniqueness, monotonicity, sign theorem sandwich theorem. Arithmetic of limits and indeterminate forms. Limits and subsequences. Limit of monotonic sequences and the Neper number. Ratio test for sequences. Fundamental limits. Infinite and infinitesimal sequences. The scale of infinite sequences. The Landau symbols.
Subsequences and their properties. Sequences and topology. The Bolzano-Weiestrass theorem. Sequential compactness. The Heine-Borel Theorem. The Cauchy condition and its equivalence with existence of finite limits.

Series: converging, diverging and irregular. Necessary condition for convergence. Geometric series, Mengoli series and telescopic series. Comparison and asymptotic comparison for series eventually of constant sign. Harmonic series. Ratio and root test; examples. Signed series: absolute convergence and convergence. Leibniz theorem.

Generalities on functions of one real variable: domain, image, injectivity, surjectivity and invertibility. Monotonicity and boundedness. Elementary functions and their graphs. Transformations, symmetries and graphs.

Limits of functions: epsilon-delta definition and sequential limits. Properties and computation of limits. Landau symbols, infinitesimals and infinites.

Continuity: epsilon-delta definition and arithmetic properties. Continuity of the composition of continuous functions. Continuity and sequential continuity. Classification of discontinuities. Continuity of piecewise defined functions and continuous extensions.
Global properties of continuous functions: Zeros, intermediate values and Weiestrass theorems. Uniform continuity and the Heine-Cantor theorem. Invertible continuous functions.

Introduction to derivatives. Differentiability and continuity. Differentiation rules: sum, product and quotient rules. Chain rule. Differentiability of inverse function.

Local maximum and minimum points, Fermat’s Theorem. Lagrange’s Theorem and monotonicity criterion. Characterization of constant functions on an interval. De L’Hospital’s Theorem. Higher order derivatives. Taylor’s formula (with Peano and Lagrange remainder). Conditions on the second derivative to determine if a stationary point is an extremum point. Convexity. Study of the graph of a function.

Introduction to the Riemann integral: definition of the integral and geometric interpretation. Necessary and sufficient conditions for integrability. Properties of the Riemann integral. Classes of integrable function. Primitives and integration rules: tables of elementaty integrals, by substitution, by parts, integration of rational functions. Integral mean theorem and fundamental theorem of calculus. Generalized integrals and integral functions.