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 7 of 7: Verify
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f(x)=0 or f(x)={0,x≥02x,x<0f(x)=0\text{ or }f(x)=\begin{cases}0,&x\ge0\\2x,&x<0\end{cases}
Detailed analysis

Direct substitution into f(xf(y))+yf(x)=xf(y)+f(xy)f(xf(y))+yf(x)=xf(y)+f(xy), separating the signs of x,yx,y, verifies both functions. These are all solutions.