Perfect level sets in many directions (Q1101561)

The author says that a real function f defined on \(<0,1>\) is admissible iff there is a finite subset A of (0,1) such that f is linear on any interval (a,b), where a,b\(\in A\) and \((a,b)\cap A=\emptyset\) and \(f(a)<\liminf_{t\to a}f(t)\) for any \(a\in A\). Then the following lemma holds: For any \(\epsilon >0\) and any (t,s), where -t and s are admissible, \(t\leq s\) on \(<0,1>\) and \(t<s\) except on a finite subset of (0,1), there exists a couple \((t^*,s^*)\), where \(-t^*\) and \(s^*\) are admissible such that: \(1.\quad t\leq t^*\leq s^*\leq s\) on \(<0,1>\); 2. \(t^*<s^*\) except on a finite subset of (0,1); 3. \(s^*- t^*<\epsilon\) on \(<0,1>\) and 4. for any continuous function f defined on \(<0,1>\) for which \(t^*\leq f\leq s^*\), for each \(x\in <0,1>\) there exists a \(y\in <0,1>\) such that \(0<| x-y| <\epsilon\) and \(f(x)=f(y).\) If \(a\in R\) and f is a real function defined on \(<0,1>\), then f is locally recurrent in the direction a iff f-a id, where id is the identity function on \(<0,1>\), is locally recurrent, that means \(\{x\in <0,1>: f(x)-ax=c\}\) is a perfect set for any \(c\in R\). For an arbitrary countable subset A of R, by means of induction - applying the lemma - the author constructs a continuous function defined on \(<0,1>\) which is locally recurrent in any direction a of A. The author also proved that any continuous function defined on \(<0,1>\) can be expressed as the sum of two locally recurrent continuous functions.