Problem 2

Find all functions

f:\mathbb R\to\mathbb R

such that

f(x^2+f(y))=y+f(x)^2

for all real

x,y

Step 3 of 5: Derive additivity for positive increments

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In plain words

Derive additivity for positive increments

f(f(x))=x,\qquad f(x^2)=f(x)^2,\qquad f(x^2+y)=f(x)+f(y)

Detailed analysis

With $t=0$ , the relations become $f(f(x))=x$ and $f(x^2)=f(x)^2$ . Given any $y$ , put $z=f(y)$ in the original equation; since $y=f(z)$ , this gives $f(x^2+y)=f(y)+f(x)^2=f(y)+f(x^2)$ . Thus $f(u+y)=f(u)+f(y)$ for every $u>0$ , writing $u=x^2$ .