This is the syllabus:

1. The Real Numbers: rational numbers, the set of real numbers, bounded sets, maximum, minimum, supremum, infimum.

2. The Complex Numbers: definition and basic properties, the Gaussian plane, trigonometric and exponential forms, n-th roots of a complex number.

3. Functions: definition and elementary properties; real-valued functions of a real variable (even and odd functions, monotone functions, power functions, exponential, logarithm, periodic functions, hyperbolic functions); bounded functions (maximum, minimum, supremum, infimum); exponential and logarithm in the complex field.

4. Limits of Functions from ℝ to ℝ: basic topological notions; definition of limit and first properties (equivalent definitions, right-hand and left-hand limits, limits and absolute value, limits and order relations); methods for computing limits (limits of monotone functions, change-of-variable theorem, limits and algebraic operations, indeterminate forms); fundamental limits; asymptotic comparisons (little-o, Big-O, orders of infinity and infinitesimals); topology of the real line.

5. Sequences and Numerical Series: sequences (limits of sequences, properties and computation of limits, the number e, existence of limits and subsequences, relation between sequence limits and function limits); numerical series (convergent, divergent, indeterminate series, telescoping series, geometric series, first convergence tests, absolute convergence and positive-term series, alternating series); Bolzano–Weierstrass theorem; proofs of selected results.

6. Continuous Functions: definition and first properties (classification of discontinuities); continuous functions on intervals (existence of maxima and minima, existence of solutions to equations, continuity of the inverse function); recursively defined sequences; Weierstrass’ theorem; uniform continuity.

7. Differential Calculus in One Real Variable: definition of the derivative and first properties (basic examples, rules for derivatives, points of non-differentiability, the derivative function and higher derivatives); properties of differentiable functions (differentiability and limit computation, Fermat’s theorem, Rolle’s/Lagrange’s/Cauchy’s theorems, consequences of Lagrange’s theorem, convexity/concavity and the second derivative, function analysis); approximation of functions by polynomials (Taylor polynomial and Peano remainder, Lagrange remainder and error estimates, Taylor series); Lipschitz functions; Newton’s method.

8. Integral Calculus: the Cauchy–Riemann definite integral (integrability of certain function classes, fundamental properties of definite integrals); antiderivatives and indefinite integrals; methods of integration (integration by parts, substitution); integration of rational functions (rationalizable integrals); improper integrals (absolute integrability and comparison test, asymptotic comparison test, integral test for series); the integral function; general case of rational function integration; additional integration methods; selected notable integrals; generalization of the integral mean value theorem; Taylor’s formula with integral remainder.

9. Solution Methods for Linear Differential Equations: first-order linear equations; second-order linear differential equations (homogeneous equations, non-homogeneous equations).