Andriyuk O. Functions defined on one-dimensional manifolds

The thesis deals with obtaining of conditions of topological equivalence of functions defined on one-dimensional manifolds and the problem of an extension of functions on a circle to the interior of disk with fixed peculiarities. The conditions are found when two continuous functions with finite number of extremal points defined on a circle (segment) are topological equivalent and the sufficient conditions which guarantee an extension of such function to the interior of disk without critical points. The problem of realization of continuous and smooth functions with arbitrary number of local extremal points on a circle by height functions is solved. The conditions are given to realize a continuous function on a circle by graph which characterize this function up to topological equivalence relation. The invariant is constructed which describes up to the topological equivalence smooth functions in the unit disk with finite number of saddles.