Question 1
Call a real-valued function $f$ *very convex* if
\[ \frac{f(x)+f(y)}{2}
\ge f\left( \frac{x+y}{2} \right) + \left\lvert x-y \right\rvert \]
holds for all real numbers $x$ and $y$.
Prove that no very convex function exists.
\[ \frac{f(x)+f(y)}{2}
\ge f\left( \frac{x+y}{2} \right) + \left\lvert x-y \right\rvert \]
holds for all real numbers $x$ and $y$.
Prove that no very convex function exists.