A first return characterization for Baire one functions (Q1332588)

Let \(B_s(y)= \{x\in [0, 1]: |x- y|< s\}\) and let \((x_n)\) be a dense (in \([0, 1]\)) sequence of distinct points of \([0, 1]\). Denote by \(r(B_s(y))\) the first element of the trajectory \((x_n)\) in \(B_s(y)\). The first return route to \(y\), \(R_y= (y_k)^\infty_{k= 1}\) is defined recursively via: \(y_1= x_0\), \(y_{k+ 1}= r(B_{|y- y_k|}(y))\) if \(y\neq y_k\), or \(y_k\) if \(y= y_k\). It is proved that a function \(f: [0, 1]\to \mathbb{R}\) is of Baire 1 class if and only if for each \(y\in [0, 1]\) we have \(\lim_{k\to \infty} f(y_k)= f(y)\) for some trajectory \((x_n)\).