We prove an analog of the Morse theorem in the case where the critical point belongs to the boundary of an $n$-dimensional manifold and find the least number of critical points for the Morse functions defined on the surfaces whose critical points coincide with the critical points of their restriction to boundary.
Citation Example:Hladysh B. I., Prishlyak A. O. Functions with nondegenerate critical points on the boundary of the surface // Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 28-37.