Use the domain and strategic substitutions to find all functions. Begin at the first unfamiliar idea; the prerequisite links help you find a shorter route. You can open any lesson.

Unsure where to start? Try the starting-point check.

  1. E01

    Functions, inverses and composition

    Use function properties precisely without assuming a formula.

    Inside: Domain, codomain and range · Composition and inverses · Injective and surjective functions · Continuity as an explicit extra condition

  2. E02

    Solving functional equations

    Find all admissible functions and verify the complete family in the original equation.

    Inside: Legal substitutions · Isolating variables · Strategic values such as zero · Injectivity and surjectivity in proofs · Integer domains and induction · Polynomial ansatz only when justified · Iteration and recurrence · Additive equations and regularity · Fixed-point methods · Candidate discovery, uniqueness and verification

These original lessons introduce the methods and give practice with solutions. A single short session is not a full assessment of Olympiad readiness. Build depth through the written challenges and your country’s official past papers.