Problem 5

Let

\mathbb R

denote the set of all real numbers. Find all functions

f:\mathbb R\to\mathbb R

such that (i) only finitely many

s\in\mathbb R

satisfy

f(s)=0

, and (ii)

f(x^4+y)=x^3f(x)+f(f(y))

for all

x,y\in\mathbb R

Step 2 of 3: Prove additivity

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f(x^4+y)=f(x^4)+f(y)

Detailed analysis

Substitution of $f(f(y))=f(y)$ gives $f(x^4+y)=x^3f(x)+f(y)$ . Setting $y=0$ gives $f(x^4)=x^3f(x)$ , hence $f(u+v)=f(u)+f(v)$ whenever $u=x^4\ge0$ . Comparing $x$ and $-x$ in the same equation gives $f(-x)=-f(x)$ for $x\ne0$ , so additivity extends to all real $u,v$ .