How does concavity look like for non-differentiable or discontinuous function?

Question

In my university textbook, it shows that if a function is concave downwards then it's graph looks something like this

https://en.m.wikipedia.org/wiki/File:ConcaveDef.png

But what if a function is discontinuous or continuous but non-differentiable. How does the graph look like then?

Or rather is there a concept concavity for discontinuous and non- differentiable function?

GEdgar · Answer 1 · 2019-05-01T12:29:39.270

3

A concave function defined on an open interval in $\mathbb R$ is continuous there. See "Every Convex Function is Continuous"

A concave function may be non-differentiable. But only at countably many points. It is right-differentiable and left-differentiable.

edited May 01 '19 at 12:29

answered May 01 '19 at 12:23

GEdgar

117,296

score 2 · Accepted Answer · answered May 01 '19 at 12:23

2

A concave function can be non-differentiable at some points. At such a point, its graph will have a corner, with different limits of the derivative from the left and right:

A concave function can be discontinuous only at an endpoint of the interval of definition.

answered May 01 '19 at 12:23

Robert Israel

470,583

Just out of curiousity why only endpoints? – William May 01 '19 at 12:25
Also by symmetry same goes for concave upwards function? – William May 01 '19 at 16:24
(1) see Gerald Edgar's link for a proof. (2) Yes. – Robert Israel May 01 '19 at 18:13
@William yes it can be proven that the function can't be discontinuous : https://pages.wustl.edu/files/pages/imce/nachbar/concavity.pdf – Marine Galantin Jun 12 '20 at 12:14

How does concavity look like for non-differentiable or discontinuous function?

2 Answers2