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# Cool functions

Find all functions $$f:\mathbb{R} \rightarrow \mathbb{R}$$ such that

$f(f(x)+y)=x+f(f(y))$

for all real numbers $$x$$ and $$y$$.

Note by Sharky Kesa
1 year, 8 months ago

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Let $$P(x,y)$$ be the statement $$f(f(x)+y) = x+f(f(y))$$.

$$P(x,0)$$ implies $$f(f(x)) = x + f(f(0))$$. Applying this to $$P$$ gives $$f(f(x)+y) = x+y+f(f(0))$$.

$$P(x,f(0))$$ implies $$f(f(x)+f(0)) = x+f(0)+f(f(0))$$. $$P(0,f(x))$$ implies $$f(f(0)+f(x)) = f(x)+f(f(0))$$. Equating the two gives $$x+f(0)+f(f(0)) = f(x)+f(f(0))$$, or $$f(x) = x + c$$ for some fixed real number $$c$$ for all real $$x$$. It can be easily verified to satisfy the equation. · 1 year, 8 months ago