How many Convex and Odd functions exist? During lunch this question popped on mind.
How many odd and convex functions $f: R \to R $ exist ?
I guessed just 1, namely $f(x)=ax$  where $a$ is a fixed real number. But I couldn't either prove it or provide counter example. 

Scalar multiples of $x$ are the only examples.
To see why, suppose you're given $(1, f(1))$ as well as $(0, 0) = (0, f(0))$. Draw the line connecting these two points, as well as its extension through $(-1, f(-1))$. 
Since $f$ is convex, its graph for $0 < x < 1$ lies on or below this line. If the graph ever dips below the line, then using the oddness of $f$ implies that it gets above the line connecting $-1$ and $0$, contradicting convexity. 
Hence, $f$ must agree with a straight line for $x \in (0, 1)$. It's not hard to extend this argument to $\mathbb{R}$. 
