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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?

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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.

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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:

enter image description here

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

enter image description here

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  • $\begingroup$ Just out of curiousity why only endpoints? $\endgroup$
    – William
    May 1, 2019 at 12:25
  • $\begingroup$ Also by symmetry same goes for concave upwards function? $\endgroup$
    – William
    May 1, 2019 at 16:24
  • $\begingroup$ (1) see Gerald Edgar's link for a proof. (2) Yes. $\endgroup$ May 1, 2019 at 18:13
  • $\begingroup$ @William yes it can be proven that the function can't be discontinuous : pages.wustl.edu/files/pages/imce/nachbar/concavity.pdf $\endgroup$ Jun 12, 2020 at 12:14

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