Some results concerning functions that preserve Baire spaces in the context of images and preimages are obtained in {\em Section IV} of [R. C. Haworth, R. A. McCoy, {\em Baire spaces}, Dissertationes Math., {\bf 151}, PWN, Warszawa 1977]. In Section 2 of the present paper further results in this direction are offered. In Sections 3 and 4 a few theorems related to some other problems on baireness of topological spaces are proposed.