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Function with discontinuity in every point

Source: Romanian District Olympiad 2006, Grade 11, Problem 4

March 11, 2006
functionreal analysisreal analysis unsolved

Problem Statement

We say that a function f:R→Rf: \mathbb R \to \mathbb R has the property (P)(P) if, for any real numbers xx, sup⁡t≤xf(x)=x. \sup_{t\leq x} f(x) = x. a) Give an example of a function with property (P)(P) which has a discontinuity in every real point. b) Prove that if ff is continuous and satisfies (P)(P) then f(x)=xf(x) = x, for all x∈Rx\in \mathbb R.
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