We discuss smooth functions which are Morse on preimages of values not being local extrema. We call such a function internally Morse or I-Morse. The Reeb graph of a smooth function is the space of all connected components of preimages of single points of it topologized with the natural quotient topology of the manifolds and a vertex of it is a point corresponding to a preimage with critical points. A smooth function is neat with respect to the Reeb graph or N-Reeb if the preimages of the vertices are the closed subsets in the manifolds of the domains with interiors being empty. We discuss I-Morse and N-Reeb functions, IN-Morse-Reeb functions. Our main result presents an IN-Morse-Reeb function respecting two such functions, on a connected sum of these given manifolds.