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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)
Problem 397

    Find the inverse of \(\displaystyle f(x)=\frac{2x-3}{3x-2}\) on \(\displaystyle \R\setminus\{\tfrac23\}\).

    Difficulty: 1.


Problem 407

    Does there exist a function \(\displaystyle f:(0,1)\to \R\) which is bounded, but has no maximum?

    Difficulty: 1.


Problem 386

    Show that the following functions are injective on the given set \(\displaystyle H\), and calculate the inverse.

     (1)  \(\displaystyle f(x)=3x-7, \ H=\R\);     (2)  \(\displaystyle f(x)=x^{2}+3x-6, \ H=[-3/2,\infty)\).

    Difficulty: 2.


Problem 387

    Show that the following functions are injective on the given set \(\displaystyle H\), and calculate the inverse.

     (1)  \(\displaystyle f(x)=\frac{x}{x+1}, \ H=[-1,1]\);     (2)  \(\displaystyle f(x)=\frac{x}{|x|+1}, \ H=\R.\)

    Difficulty: 2. Solution is available for this problem.


Problem 398

    Are the following functions injective on \(\displaystyle [-1,1]\)?

    a) \(\displaystyle \displaystyle f(x)=\frac{x}{x^2+1}\),    b) \(\displaystyle \displaystyle g(x)=\frac{x^{2}}{x^2+1}\).

    Difficulty: 2. Solution is available for this problem.


Problem 401

    Prove that all function \(\displaystyle f:\R \to \R\) can be obtained as the sum of an even and an odd function.

    Difficulty: 2.


Problem 403

    Let \(\displaystyle f(x)=x^3\) if \(\displaystyle x\) is rational, and \(\displaystyle f(x)=-x^3\) if \(\displaystyle x\) is irrational. Does \(\displaystyle f(x)\) have a unique inverse on \(\displaystyle (-\infty,+\infty )\)?

    Difficulty: 2.


Problem 406

    Prove that if \(\displaystyle f\) is strictly convex on the interval \(\displaystyle I\), then every line intersects the graph of \(\displaystyle f\) in at most 2 points.

    Difficulty: 2.


Problem 408

    Does there exist a function \(\displaystyle f:[0,1]\to \R\) which is bounded, but has no maximum?

    Difficulty: 2.


Problem 412

    Prove that if \(\displaystyle a_{1},...,a_{n}\geq 0\) and \(\displaystyle k>1\) is an integer, then \(\displaystyle \frac{a_{1}+...+a_{n}}{n}\leq \sqrt[k]{\frac{a_{1}^{k}+...+a_{n}^{k}}{n}}\).

    Difficulty: 3.


Problem 389

    Construct a non-constant periodic function with arbitrarily small periods.

    Difficulty: 4.


Problem 405

    Let \(\displaystyle f(x)=\max\{x,1-x,2x-3\}\). Is it monotone, or convex?

    Difficulty: 4.


Problem 409

    Does there exist a monotone function \(\displaystyle f\) such that

     (1)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=(0,1);\phantom{\,\cup [2,3]}\)      (2)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1]\cup [2,3]\);
     (3)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1)\cup [2,3]\);      (4)  \(\displaystyle D(f)=[0,1]\), \(\displaystyle R(f)=[0,1)\cup (2,3]\)?

    Difficulty: 4. Solution is available for this problem.


Problem 414

    Prove that if \(\displaystyle g:A\to B\) and \(\displaystyle f:B\to C\) are convex, and \(\displaystyle f\) is monotone increasing, then \(\displaystyle f\circ g\) is convex.

    Difficulty: 4.


Problem 415

    Prove that if \(\displaystyle f\) is convex, then it can be obtained as the sum of a monotone increasing and a monotone decreasing function.

    Difficulty: 4.


Problem 411

    Prove that \(\displaystyle x^{k}\) is strictly convex on \(\displaystyle [0,\infty)\), for all \(\displaystyle k>1\) integer.

    Difficulty: 5.


Problem 388

    Find a function \(\displaystyle f:[-1,1]\to[-1,1]\) such that \(\displaystyle f(f(x))=-x \forall x\in[-1,1]\).

    Difficulty: 7. Solution is available for this problem.


Problem 417

    Can we obtain the function \(\displaystyle x^2\) as a sum of two periodic functions?

    Difficulty: 7.


Problem 410

    Does there exist a function which attains every real values on any interval?

    Difficulty: 8.


Problem 418

    Can we obtain the function \(\displaystyle x^2\) as a sum of three periodic functions?

    Difficulty: 10.


        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)
        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)
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