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