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We say that the real-valued function

f(x)

defined on the interval

(0,1)

is approximately continuous on

(0,1)

if for any

x_0 \in (0,1)

and

\varepsilon >0

the point

x_0

is a point of interior density

1

of the set H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}. Let

F \subset (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

Problem Statement