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Is there such a function $f:\mathbb R\rightarrow\mathbb R$, that for any real $x$ and $y$, we have the equality: $$ \frac{f(x)+f(y)}{2}=f\left({\frac{x+y}{2}}\right)+|x+y|\;\;\;? $$

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Suppose $f$ satisfies the given equation for any pair $(x,y) \in \mathbb R^2$. Then for $x = y$ we get:

$$ \frac{f(x) + f(x)}{2} = f \left( \frac{x + x}{2} \right) + \left| x + x \right| \iff f(x) = f(x) + \left| 2x \right| \iff x = 0 $$

So $f$ cannot be satisfied for any pair $(x,x)$ with $x \neq 0$. This is a contradiction and hence such an $f$ cannot exist.

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There is no such function.

Indeed for $x=y=1$, you would have : $$\dfrac{f(1)+f(1)}{2}=f\left(\dfrac{1+1}{2}\right) + |1+1| \Rightarrow 0=2$$

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