We define a real-valued function to be nice, if
$\dfrac{f(x)+f(y)}{2} \ge f\bigg(\dfrac{x+y}{2}\bigg) + |x - y|$
holds for any $x,y \in \mathbb{R}$. Prove that there doesn't exist any nice functions.
My Work:
Let $y=-x$. Then we have, $\dfrac{f(x)+f(-x)}{2} \ge f(0) +2x$
Similarly for $x=y=0$ we have, $f(0) \ge f(0)$
I stopped because I think I am going in a completely wrong direction. Help would be greatly appreciated. Thank you
Let $g(x)=f(x)+f(-x)-2f(0)$. From your observation it follows that $g(x) \geq 2f(0)+4x-2f(0)=4x \geq 0$ for any $x \geq 0$.
Now check that $\frac {g(x)+g(y)} 2 \geq g(\frac {x+y} 2)+|x-y|$ and note that $g(0)=0$ . Put $y=0$ to get $g(x)\geq 2g(\frac x 2) +2|x|$ for any $x \geq 0$. Iteration of this gives $g(x)\geq 2^{n}g(\frac x {2^{n}})+2^{n}|x|$. The first term is non-negative (for $x \geq 0$) so $g(x)\geq 2^{n}|x|$ . You get a contradiction by letting $n \to \infty$.