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 3 of 3: Use finite zeros and conclude

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Z=\{s\in\mathbb R:f(s)=0\}=\{0\}\implies f\text{ injective}

Detailed analysis

If $h\ne0$ and $f(h)=0$ , additivity gives $f(nh)=0$ for every positive integer $n$ , contradicting finiteness. Hence the zero set is $Z=\{0\}$ , and $f(a)=f(b)$ implies $f(a-b)=0$ , so $f$ is injective. Since $f(f(x))=f(x)$ , injectivity gives $f(x)=x$ . Direct substitution verifies the identity function.