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Real Analysis 1-4 (semesters 1-4)
    Basic notions. Axioms of the real numbers (semester 1, weeks 1-2; 0 problems)
        Fundaments of Logic (semester 1, week 1; 16 problems)
        Proving Techniques: Proof by Contradiction, Induction (semester 1, week 1; 20 problems)
            Fibonacci Numbers (semester 1, week 1; 6 problems)
        Solving Inequalities and Optimization Problems by Inequalities between Means (semester 1, week 1; 19 problems)
        Sets, Functions, Combinatorics (semester 1, week 2; 22 problems)
    Axioms of the real numbers (semester 1, weeks 2-4; 0 problems)
        Field Axioms (semester 1, week 2; 5 problems)
        Ordering Axioms (semester 1, week 2; 6 problems)
        The Archimedean Axiom (semester 1, week 2; 3 problems)
        Cantor Axiom (semester 1, week 2; 7 problems)
        The Real Line, Intervals (semester 1, week 3; 15 problems)
        Completeness Theorem, Connectivity, Topology of the Real Line. (semester 1, week 3; 4 problems)
        Powers (semester 1, week 4; 3 problems)
    Convergence of Sequences (semester 1, weeks 4-6; 0 problems)
        Theoretical Exercises (semester 1, week 4; 61 problems)
        Order of Sequences, Threshold Index (semester 1, week 4; 22 problems)
        Limit Points, liminf, limsup (semester 1, week 5; 15 problems)
        Calculating the Limit of Sequences (semester 1, week 5; 33 problems)
        Recursively Defined Sequences (semester 1, week 5; 21 problems)
        The Number $e$ (semester 1, week 5; 13 problems)
        Bolzano–Weierstrass Theorem and Cauchy Criterion (semester 1, week 6; 5 problems)
        Infinite Sums: Introduction (semester 1, week 6; 21 problems)
    Cardinalities of Sets (semester 1, week 7; 0 problems)
        Countable and not countable sets (semester 1, week 7; 6 problems)
        Not countable Sets (semester 1, week 7; 0 problems)
    Limit and Continuity of Real Functions (semester 1, weeks 7-10; 0 problems)
        Global Properties of Real Functions (semester 1, week 7; 20 problems)
        Continuity and Limits of Functions (semester 1, week 8; 33 problems)
        Calculating Limits of Functions (semester 1, week 8; 29 problems)
        Continuity and Convergent Sequences (semester 1, week 9; 0 problems)
        Continuous Functions on a Closed Bounded Interval (semester 1, week 9; 9 problems)
        Uniformly Continuous Functions (semester 1, week 10; 5 problems)
        Monotonity and Continuity (semester 1, week 10; 2 problems)
        Convexity and Continuity (semester 1, week 10; 7 problems)
    Elementary functions (semester 1, weeks 11-12; 0 problems)
        Arclength of the Graph of the Function (semester 1, week 11; 0 problems)
        Exponential, Logarithm, and Power Functions (semester 1, week 11; 17 problems)
        Inequalities (semester 1, week 12; 1 problems)
        Trigonometric Functions and their Inverses (semester 1, week 12; 3 problems)
    Differential Calculus and its Applications (semester 2, weeks 0-3; 0 problems)
        The Notion of Differentiation (semester 2, week 0; 47 problems)
        Tangents (semester 2, week 1; 11 problems)
        Higher Order Derivatives (semester 2, week 1; 13 problems)
        Local Properties and the Derivative (semester 2, week 1; 4 problems)
        Mean Value Theorems (semester 2, week 1; 3 problems)
Problem 759

    Using the Lagrange mean value theorem prove that if \(\displaystyle f\) is differentiable on \(\displaystyle \R\) and \(\displaystyle f'\) is bounded, then \(\displaystyle f\) is Lipschitz.

    Difficulty: 4.


Problem 762

    Using the Lagrange mean value theorem prove that if \(\displaystyle f'(a+0)\) exists, then \(\displaystyle f'_+(a)\) also exists and they are equal.

    Difficulty: 5.


Problem 770

    Let \(\displaystyle a_1<a_2<\ldots<a_n\) and \(\displaystyle b_1<b_2<\ldots<b_n\) be real numbers Show that

    \(\displaystyle \det\begin{pmatrix} e^{a_1b_1} & e^{a_1b_2} & \dots & e^{a_1b_n} \\ e^{a_2b_1} & e^{a_2b_2} & \dots & e^{a_2b_n} \\ \vdots & \vdots & \ddots & \vdots \\ e^{a_nb_1} & e^{a_nb_2} & \dots & e^{a_nb_n} \\ \end{pmatrix} >0. \)

    (KöMaL A. 463., October 2008)

    Difficulty: 9. Solution is available for this problem.


        Number of Roots (semester 2, week 1; 5 problems)
        Exercises for Extremal Values (semester 2, week 2; 2 problems)
            Inequalities, Estimates (semester 2, week 2; 14 problems)
        The L'Hospital Rule (semester 2, week 2; 14 problems)
        Polynomial Approximation, Taylor Polynomial (semester 2, week 3; 20 problems)
        Convexity (semester 2, week 3; 5 problems)
        Analysis of Differentiable Functions (semester 2, week 3; 6 problems)
    Riemann Integral (semester 2, weeks 4-11; 0 problems)
        Definite Integral (semester 2, week 4; 11 problems)
        Indefinite Integral (semester 2, weeks 5-6; 13 problems)
        Properties of the Derivative (semester 2, week 6; 2 problems)
        Newton-Leibniz formula (semester 2, week 6; 2 problems)
        Integral Calculus (semester 2, week 7; 5 problems)
        Applications of the Integral Calculus (semester 2, week 8; 4 problems)
            Calculating the Area and the Volume (semester 2, week 8; 0 problems)
            Calculating the Arclength (semester 2, week 8; 3 problems)
            Surface Area of Surfaces of Revolution (semester 2, week 8; 0 problems)
        Integral and Inequalities (semester 2, week 8; 5 problems)
        Improper Integral (semester 2, week 9; 9 problems)
        Liouville Theorem (semester 2, week 10; 0 problems)
        Functions of Bounded Variation (semester 2, week 11; 2 problems)
        Riemann-Stieltjes integral (semester 2, week 11; 2 problems)
    Infinite Series (semester 2, weeks 12-13; 38 problems)
    Sequences and Series of Functions (semester 3, weeks 1-2; 0 problems)
        Convergence of Dequences of Functions (semester 3, week 1; 16 problems)
        Convergence of Series of Functions (semester 3, week 1; 17 problems)
        Taylor and Power Series (semester 3, week 2; 12 problems)
    Differentiability in Higher Dimensions (semester 3, weeks 3-6; 0 problems)
        Topology of the $n$-dimensional Space (semester 3, week 3; 29 problems)
        Real Valued Functions of Several Variables (semester 3, weeks 3-4; 0 problems)
            Limits and Continuity in $R^n$ (semester 3, week 3; 16 problems)
            Differentiation in $R^n$ (semester 3, week 4; 62 problems)
        Vector Valued Functions of Several Variables (semester 3, weeks 5-6; 0 problems)
            Limit and Continuity (semester 3, week 5; 3 problems)
            Differentiation (semester 3, week 5; 11 problems)
            Implicite functions (semester 3, week 6; 0 problems)
    Jordan Measure and Riemann Integral in Higher Dimensions (semester 3, weeks 7-9; 60 problems)
    Integral Theorems of Vector Calculus (semester 3, week 10 -- semester 4, week 1; 0 problems)
        The Line Integral (semester 3, week 10; 11 problems)
        Newton-Leibniz Formula (semester 3, week 11; 6 problems)
        Existence of the Primitive Function (semester 3, week 12; 13 problems)
        Integral Theorems in 2D (semester 4, week 1; 2 problems)
        Integral Theorems in 3D (semester 4, week 1; 12 problems)
    Measure Theory (semester 4, weeks 3-99; 0 problems)
        Set Algebras (semester 4, week 3; 9 problems)
        Measures and Outer Measures (semester 4, week 4; 8 problems)
        Measurable Functions. Integral (semester 4, week 5; 10 problems)
        Integrating Sequences and Series of Functions (semester 4, weeks 7-8; 11 problems)
        Fubini Theorem (semester 4, week 9; 1 problems)
        Differentiation (semester 4, weeks 11-12; 7 problems)
Complex Analysis (semester 5)
    Complex differentiability (semester 5, week 0; 0 problems)
        Complex numbers (semester 5, week 0; 21 problems)
            The Riemann sphere (semester 5, week 0; 1 problems)
    Regular functions (semester 5, weeks 1-2; 0 problems)
        Complex differentiability (semester 5, week 1; 7 problems)
        The Cauchy-Riemann equations (semester 5, week 1; 3 problems)
        Power series (semester 5, weeks 1-2; 0 problems)
            Domain of convergence (semester 5, week 2; 9 problems)
            Regularity of power series (semester 5, week 2; 2 problems)
            Taylor series (semester 5, week 2; 1 problems)
        Elementary functions (semester 5, week 2; 0 problems)
            The complex exponential and trigonometric functions (semester 5, week 2; 8 problems)
            Complex logarithm (semester 5, week 2; 12 problems)
    Complex Line Integral and Applications (semester 5, weeks 3-5; 0 problems)
        The complex line integral (semester 5, week 3; 9 problems)
        Cauchy's theorem (semester 5, week 3; 6 problems)
        The Cauchy formula (semester 5, week 4; 12 problems)
        Power and Laurent series expansions (semester 5, week 5; 0 problems)
            Power series expansion (semester 5, week 5; 2 problems)
            Liouville's Theorem (semester 5, week 5; 7 problems)
        Local properties of holomorphic functions (semester 5, week 5; 0 problems)
        Consequences of analyticity (semester 5, week 5; 8 problems)
            The maximum principle (semester 5, week 5; 4 problems)
        Laurent series (semester 5, week 5; 9 problems)
    Isolated singularities (semester 5, weeks 5-8; 0 problems)
        Singularities (semester 5, week 5; 4 problems)
        Cauchy's theorem on residues (semester 5, weeks 7-8; 15 problems)
            Residue calculus (semester 5, week 7; 6 problems)
            Applications (semester 5, week 7; 0 problems)
                Evaluation of series (semester 5, week 7; 7 problems)
                Evaluation of integrals (semester 5, week 7; 26 problems)
            The argument principle and Rouche's theorem (semester 5, week 8; 7 problems)
    Conformal maps (semester 5, weeks 9-10; 0 problems)
        Fractional linear transformations (semester 5, week 9; 20 problems)
        Riemann mapping theorem (semester 5, week 9; 11 problems)
        Schwarz lemma (semester 5, week 9; 12 problems)
        Caratheodory's theorem (semester 5, week 9; 2 problems)
        Schwarz reflection principle (semester 5, week 10; 2 problems)
    Harmonic functions (semester 5, weeks 11-12; 8 problems)
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government