Is this function a convex function?

Asked Nov 09 '20 at 05:39

Active Nov 10 '20 at 16:12

Viewed 22 times

\begin{equation} [ f(x) = \begin{cases} x^2 & \quad x>0\\ 1/2 & \quad x=0 \end{cases} ] \end{equation}

Is this function convex? Is the epigraph of this function a convex set?

I think this function is convex in $[0,+\infty)$, but I do not think the epigraph of this function a convex set. I'm really confused.

asked Nov 09 '20 at 05:39

Mike Mathcook

1

If you think that the epigraph is not a convex set, then you're saying that there are a pair of points $A$ and $B$ where the line segment $\overline{AB}$ is not contained in the epigraph. Do you really think you can find a pair of points like this? – Eric Perkerson Nov 09 '20 at 06:04
A = (0,0), B = (0,1/2), Is the line segment $\overline{AB}$ contained in the epigraph? – Mike Mathcook Nov 09 '20 at 06:28
1

Is $(0, 0)$ a point in the epigraph though? It isn't above a point on the graph. – Eric Perkerson Nov 09 '20 at 06:32
I got it! Thank you for your great answer! – Mike Mathcook Nov 09 '20 at 07:00

0 Answers0