MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that for all x,y,z∈Rx,y,z\in\mathbb R, f(f(x)+f(y)+f(z))=f(f(x)−f(y))+f(2xy+f(z))+2f(xz−yz)f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).
Step 5 of 8: Fibres have two points at most
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f(x)=f(y)⟹x=±yf(x)=f(y)\Longrightarrow x=\pm y
Detailed analysis

If equal values occurred at nonzero x,yx,y with x≠±yx\ne\pm y, evenness lets us assume they have the same sign. Scaling gives f(rx)=f(x)f(rx)=f(x) for some positive r≠1r\ne1; comparing the equation at scaled arguments then forces ff to be constant, a contradiction.