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 4 of 7: Use the upper bound
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f(x)≤0f(x)\le0
Detailed analysis

A positive value in the range would produce arbitrarily large positive values, so f(x)≤0f(x)\le0 for every xx.