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This paper focuses on the problem of topological equivalence of functionswith isolated critical points on the boundary of a compact surface $M$ whichare also isolated critical points of their restrictions to the boundary. Thisclass of functions we denote by $\Omega(M)$. Firstly, we've obtained thetopological classification of above-mentioned functions in some neighborhood oftheir critical points. Secondly, we've constructed a chord diagram from theneighborhood of a critical level. Also the minimum number of critical points ofsuch functions is being considered. And finally, the criterion of globaltopological equivalence of functions which belong to $\Omega(M)$ and have threecritical points has been developed.

Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold