Function decompositions related to the Luzin $N$-property

We introduce a class of continuous completely regular functions satisfying the $N$-property. We obtain a decomposition of an arbitrary continuous function into the sum of two functions the first of which is completely regular and the second does not enjoy the $N$-property. We define a class of strongly regular Borel functions for which we prove the Luzin $N$-property. We demonstrate that the image of every Lebesgue measurable set of a strongly regular function is measurable. From an arbitrary Borel function we extract a strongly regular function and a function that does not enjoy the $N$-property.