Extensions of Darboux functions (Q752209)

The following result is proven under the assumption of the continuum hypothesis: Let f: \(F\to R^ 2\) where F is closed and convex and f is Darboux (i.e. maps arcs onto connected sets). Then for each \(\epsilon >0\) there exists a Darboux extension \(f^*:\;R^ 2\to R^ 2\) of f such that the set of continuity points of f and \(f^*\) agree and range \(f^*\) is in the \(\epsilon\)-neighborhood of range f.