Continuous Functions on the Sphere and Isometries

The theorem of Borsuk-Ulam states that n odd functions on the n-dimensional sphere always have a common zero. We have tried to obtain a similar theorem by "slightly" changing the conditions for the functions, but it turned out that only a very weak analogue can be expected in our case. Here we want to prove a few results and mention some of the questions which have remained unanswered.