MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R that are bounded above and satisfy f(xf(y))+yf(x)=xf(y)+f(xy)f(xf(y))+yf(x)=xf(y)+f(xy) for all real x,yx,y.
Step 6 of 7: Nonzero case
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f(x)=2x(x<0)f(x)=2x\quad(x<0)
Detailed analysis

If the function is not zero, choose b<0b<0 with c=f(b)<0c=f(b)<0. Then f(c)=2cf(c)=2c; substituting y=cy=c forces f(x)=2xf(x)=2x for every negative xx.