We say that the real-valued function f(x) defined on the interval (0,1) is approximately continuous on (0,1) if for any x0∈(0,1) and ε>0 the point x0 is a point of interior density 1 of the set H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}. Let F⊂(0,1) be a countable closed set, and g(x) a real-valued function defined on F. Prove the existence of an approximately continuous function f(x) defined on (0,1) such that f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ .
M. Laczkovich, Gy. Petruska functiontopologyreal analysisreal analysis unsolved