Every almost continuous function is polygonally almost continuous. (Q1852323)

A function \(f: \mathbb{I} \rightarrow \mathbb{R}\) is almost continuous (AC) if whenever \(U\subset \mathbb{I} \times \mathbb{R}\) is an open set containing the graph of \(f\), then \(U\) contains the graph of a continuous function. A function \(f: \mathbb{I} \rightarrow \mathbb{R}\) is polygonally almost continuous (PAC) if whenever \(U\subset \mathbb{I} \times \mathbb{R}\) is an open set containing the graph of \(f\), then \(U\) contains the graph of a picewise linear continuous function with all vertices of \(f\). The main result of this paper: Every AC function \(f: \mathbb{I} \rightarrow \mathbb{R}\) is PAC.