How can I show that the below function is convex in $x$ and convex in $e$? $x_1 , x_2\geq 0$ and $e$ is a random variable which has a limited expectation.
\begin{equation} Q(x,e)=\begin{cases} 1-x_1 & 0\leq e <x_1+x_2\\ e+1-2x_1-x_2 & x_1+x_2 \leq e <1+x_2\\ 2(e-x_1-x_2) & 1+x_2\leq e \end{cases} \end{equation}
It seems like there is an error in the function definition. Take $x_1=e=2$ fixed, then the function simplifies to: $$f(x) = Q((2,x),2) = \begin{cases}-1 & 0 \leq 2 < 2+x \\ -1-x & 2+x \leq 2 < 1+x \\ -2x & 1+x \leq 2 \end{cases}$$ The second case can never occur, and the other two cases can be simplified to: $$f(x) = \begin{cases}-1 & 0 < x \\ -2x & x \leq 1 \end{cases}$$ How do I evaluate this function at $x=0.5$?
