S. Marcus showed that there is a quasicontinuous function from the interval [0, 1] to \(\mathbb {R}\) which is not Lebesgue measurable. We prove that if X is either an uncountable Polish space or a locally pathwise connected perfectly normal topological space with at least one non isolated point, then there is a quasicontinuous non Borel measurable function from X to [0, 1]. We also found new conditions under which for every quasicontinuous function there is an equivalent Borel measurable quasicontinuous function. If X is a Baire space and Y is a separable metric space, then every Borel measurable function \(f: X \rightarrow Y\) of the first class is cliquish.

S. Marcus 证明了从区间 [0, 1] 到\(\mathbb {R}\) 之间存在一个拟连续函数，它是 Lebesgue 不可测的。我们证明，如果X是不可数波兰空间或具有至少一个非孤立点的局部路径连接的完美正则拓扑空间，则存在从X到 [0, 1]的准连续非 Borel 可测函数。我们还发现了新条件，在该条件下，对于每个拟连续函数都有一个等效的 Borel 可测拟连续函数。如果X是 Baire 空间，Y是可分度量空间，则第一类的每个 Borel 可测函数\(f: X \rightarrow Y\)都是派系。