Course

Book 3. Functional Equations

Book 3. Functional Equations

  • 1. What Is a Functional Equation?
  • 2. First Substitutions
  • 3. Linear Functional Equations
  • 4. Injectivity and Surjectivity
  • 5. Cauchy-Type Equations
  • 6. Functional Equations on Integers
  • 7. Polynomial Functional Equations
  • 8. Iteration
  • 9. Inequality Conditions in Functional Equations
  • 10. Advanced Functional Equations
  • 11. Mixed Functional Equations
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Theory Examples Practice Ladders

Chapters

Chapters

Chapter

What Is a Functional Equation?

An introductory but olympiad-level module: domains, verification, shifts to additivity, first iterations, and the difference between \(\mathbb Q\) and \(\mathbb R\).
20 Problems
Theory Examples Practice

Chapter

First Substitutions

A module on first substitutions in functional equations: \(x=0\), \(y=0\), \(x=y\), \(x=-y\), variable changes, and source-inspired regional/final-level problems.
24 Problems
Theory Examples Practice

Chapter

Linear Functional Equations

A module on proving linear and affine forms: additivity on rationals, regularity on reals, shifts, constant functions, and compositions.
20 Problems
Theory Examples Practice

Chapter

Injectivity and Surjectivity

A module on proving injectivity, using surjectivity, cancelling an outside function, and working with preimages in functional equations.
20 Problems
Theory Examples Practice

Chapter

Cauchy-Type Equations

A module on classical Cauchy-type functional equations: additivity, multiplicativity, midpoint equations, the quadratic version, and the differences between the domains \(\mathbb N\), \(\mathbb Z\), \(\mathbb Q\), and \(\mathbb R\).
20 Problems
Theory Examples Practice

Chapter

Functional Equations on Integers

A module on discrete functional equations: induction, parity, divisibility, recurrences, residue classes, and finite domains.
20 Problems
Theory Examples Practice

Chapter

Polynomial Functional Equations

A module on functional equations where the unknown function is a polynomial: degree comparison, leading coefficients, finite differences, and root arguments.
20 Problems
Theory Examples Practice

Chapter

Iteration

A module on repeated application of a function: \(f(f(x))\), fixed points, cycles, involutions, idempotents, and iteration chains.
20 Problems
Theory Examples Practice

Chapter

Inequality Conditions in Functional Equations

A module on how monotonicity, boundedness, positivity, and order preservation force functional equations to have ordinary linear solutions.
20 Problems
Theory Examples Practice

Chapter

Advanced Functional Equations

A module on mixed and parametric functional equations: additivity plus products, systems of functions, rational and real domains, compositions, and parameters.
20 Problems
Theory Examples Practice

Chapter

Mixed Functional Equations

A mixed module without a predetermined method: Cauchy, Jensen, iterations, finite sets, polynomials, inequalities, and parameters.
20 Problems
Theory Examples Practice