Exceptional Sets for Universally Polygonally Approximable Functions

In [9], the present authors and Richard O’Malley showed that in order for a function be universally polygonally approximable it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ − (1− ε) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = S∞ n=1 En, such that each Ei is closed and 1-symmetrically porous, then there is a universally polygonally approximable function, f , whose set of points of non-quasicontinuity is precisely E. Although it is tempting to call this a partial converse to our earlier theorem it might be more since it is not known if these two notions of symmetric porosity differ in the class of Fσ sets.