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 6 of 8: Determine the value at zero
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f(0)=0f(0)=0
Detailed analysis

Set z=0z=0 and replace yy by −y-y. Evenness cancels all other terms and gives f(2xy+f(0))=f(−2xy+f(0))f(2xy+f(0))=f(-2xy+f(0)). The two-point fibre property forces f(0)=0f(0)=0.