Beginner Functional Equation

$f(x+f(y))=x+y\cdots(1)$

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Put $x=0$ into $(1)$ , $f(f(y))=y\cdots(2)$

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Put $y=0$ into $(1)$ , $f(x+f(0))=x$
Apply $f$ on both sides, $f(f(x+f(0)))=f(x)$
Apply $(2)$ on LHS, $x+f(0)=f(x)$
i.e. $f(x)=x+f(0)\cdots(3)$

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It remains to find the value of $f(0)$
Apply $(3)$ on $(1)$ repeatedly, $LHS=f(x+f(y))=x+f(y)+f(0)=x+y+2f(0)$
While $RHS=x+y$
$\therefore f(0)=0\cdots(4)$

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Combining $(3)$ and $(4)$ , $f(x)=x$

$f( -f(y) + f(y)) = f(0) = -f(y) + y, \quad \forall y \in \mathbb{R} \Rightarrow$ $f(y) = y - f(0), \space \forall y \in \mathbb{R} \Rightarrow$ $x + f(y) - f(0) = f(x + f(y)) = x + y, \space \forall x,y \in \mathbb{R} \Rightarrow$ $f(y) = y + f(0) = y - f(0),\space \forall y \in \mathbb{R} \Rightarrow f(0) = 0 \Rightarrow f(y) = y, \space \forall y \in \mathbb{R}$