## Fine tuning the recoverability of Baire one functions.(English)Zbl 0854.26002

Any sequence $$\{x_n\}^\infty_{n = 0}$$ of distinct points in $$[0,1]$$ which is dense in $$[0,1]$$ is said to be a trajectory. Let $$\{x_n\}$$ be a fixed trajectory. For a given interval, or finite union of intervals, $$H \subset [0,1]$$, $$r(H)$$ will be the first element of the trajectory $$\{x_n\}$$ in $$H$$. Let $$x \in [0,1]$$, $$\rho > 0$$ and $$B_\rho(x) = \{y \in [0,1] : |x - y|< \rho\}$$. The first return route to $$x$$, $$R_x = \{y_k\}^\infty_{k = 1}$$, is defined recursively via $$y_1 = x_0$$, $$y_{k+1} = r(B_{|x- y_k|} (x))$$ if $$x \neq y_k$$ and $$y_{k+1} = y_k$$ if $$x = y_k$$. A function $$f : [0,1] \to R$$ $$(R$$ – the real line) is first return recoverable with respect to $$\{x_n\}$$ at $$x$$ provided that $$\lim_{k \to \infty} f(y_k) = f(x)$$, and if this happens for each $$x \in [0,1]$$, then $$f$$ is first return recoverable with respect to $$\{x_n\}$$. For $$0 < x \leq 1$$ the left first return path to $$x$$ based on $$\{x_n\}$$, $$P^1_x = \{t_k\}$$, is defined recursively via $$t_1 = r(0,x)$$, and $$t_{k + 1} = r(t_k, x)$$ and for $$0 \leq x < 1$$ the right first return path to $$x$$ based on $$\{x_n\}$$, $$P^r_x = \{s_k\}$$, is defined analogously. The function $$f : [0,1] \to R$$ is first return continuous at $$x \in (0,1)$$ with respect to the trajectory $$\{x_n\}$$ provided $$\lim_{t \to x, t\in P^1_x} f(t) = f(x)$$ and $$\lim_{s\to x, s\in P^r_x} f(s) = f(x)$$. Analogously, for each $$x \in [0,1]$$ the first return approach to $$x$$ based on $$\{x_n\}$$, $$A_x = \{u_k\}$$, is defined recursively via $$u_1 = r((0,1) - \{x\})$$, and $$u_{k+1} = r(B_{|x - u_k|} (x) - \{x\})$$. $$f$$ is first return approachable at $$x$$ with respect to the trajectory $$\{x_n\}$$ provided $$\lim_{u \to x, u\in A_x} f(u) = f(x)$$.
The authors categorize the points $$x$$ into types depending upon how isolated $$(x,f(x))$$ is in the graph of $$f$$: Type I points are those $$x \in (0,1)$$ for which $$(x,f(x))$$ is isolated on neither the left nor the right. Type II points are those $$x \in (0,1)$$ for which $$(x,f(x))$$ is isolated on exactly one side. The main result of the paper follows: Let $$f : [0,1] \to R$$ be a Baire one function. Then there exists a trajectory $$\{x_n\}$$ such that a) $$f$$ is first return recoverable with respect to $$\{x_n\}$$; b) If $$x$$ is a type I point, then $$f$$ is first return continuous at $$x$$ with respect to $$\{x_n\}$$; c) If $$x$$ is a type II point, then $$f$$ is first return approachable at $$x$$ with respect to $$\{x_n\}$$.

### MSC:

 26A03 Foundations: limits and generalizations, elementary topology of the line 26A21 Classification of real functions; Baire classification of sets and functions 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable