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 5 of 7: Nonnegative inputs
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f(x)=0(x≥0)f(x)=0\quad(x\ge0)
Detailed analysis

The substitutions in the official solution give f(0)=0f(0)=0 and, using the identity with a positive xx and y=1/xy=1/x, give f(x)=0f(x)=0 for all x>0x>0.