Convexity and concavity of discontinuous functions

Question

QUESTION F(x) =-x for x>=0 and F(x)=x for x<=0 Is the function convex/(strictly), concave/(strictly)

I have attempted the answer but got strictly concave but isnt a discontinuous function meant to be neither convex nor concave?

Thanks for you help in advance :)

Answer 1

What you have is a piecewise function, but in this case, it is continuous.

The idea of strict concavity is that, for any two points on the graph, the segment connecting the two points lies below the graph except at those two points. For concavity, this segment must simply lie below or on the graph at all points.

You should be able to find two points whose connecting segment meets the graph in between the endpoints, so this function is not strictly concave. However, it is, indeed, concave.

To show this, you'll need to proceed by cases, depending on the signs of $x$ and $y.$ Note that you should not specify values for $x$ or $y,$ nor should you specify a value for $\lambda.$ See what you can do, and let me know if you're still stuck.

Answer 2

Not sure if what I have done is right. and I am a bit confused what the outcome would be for case 2