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Every almost continuous function is polygonally almost continuous. (English) Zbl 1035.26005

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.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable