If f(x,y)+f(y,x)=0 for any x,y, is it true that f(x,y)=g(x)-g(y)

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Discussion Overview

The discussion centers on the functional equation f(x,y) + f(y,x) = 0 and whether it implies that f(x,y) can be expressed in the form g(x) - g(y). The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant questions if f(x,y) can be expressed as g(x) - g(y) under the given condition.
  • Another participant argues against this, providing an example of a function yg(x) - xg(y) that also satisfies the relation.
  • A later reply suggests a more general form, stating that f(x,y) can be expressed as g(x,y) - g(y,x).

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the form of f(x,y).

Contextual Notes

The discussion does not resolve the implications of the functional equation fully, and the proposed forms depend on the definitions and properties of the functions involved.

AlonsoMcLaren
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If f(x,y)+f(y,x)=0 for any x,y, is it true that f(x,y) can always be written as g(x)-g(y)?

If so, how to prove it?
 
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No. For example yg(x)-xg(y) also satisfies the relation.
Regards.
 
Thanks
 
In general f(x,y)=g(x,y)-g(y,x) obviously.
 

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