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 3 of 8: Scale equal values
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f(u1)=f(u2)⟹f(su1)=f(su2) (∀s)f(u_1)=f(u_2)\Longrightarrow f(su_1)=f(su_2)\ (\forall s)
Detailed analysis

Comparing the equation with z=u1z=u_1 and z=u2z=u_2 shows that equal function values remain equal after every common scaling of the inputs.