I am interested in conditions when the subdifferential is empty.


If $f$ is lower semicontinuous, convex, proper, then $\partial f(x) = \emptyset$ if and only if $f(x) = +\infty$

Is this true?

I know that $f(x) = +\infty \implies \partial f(x) = \emptyset$.

But is the reverse true?

$\partial f(x) = \emptyset \implies f(x) = +\infty$?

If not, is there a general condition that says when $\partial f(x)$ is empty?

  • $\begingroup$ Are you assuming that $f$ is convex? I don't think $f(x) = x^2 \sin(x)$ has a nonempty subdifferential anywhere. $\endgroup$ – David Kraemer Sep 16 at 0:18
  • $\begingroup$ @DavidKraemer Yes, every nice assumptions. lower semicontinuous, proper, convex $\endgroup$ – Sanjay Gupta Sep 16 at 0:19
  • $\begingroup$ Note that convexity implies lower semicontinuity (in fact, convex functions are locally Lipschitz). $\endgroup$ – Chris Sep 16 at 0:46
  • $\begingroup$ @Chris The convex indicator function of a bounded open set is an example of a convex function that is not lower continuous. $\endgroup$ – littleO Sep 16 at 0:51
  • $\begingroup$ Sorry, I meant real-valued, but I guess that the author of the question is specifically wondering about when the function takes the value infinity. Oops. $\endgroup$ – Chris Sep 16 at 2:07

The function $$ f(x) = \begin{cases} -\sqrt{x} &\quad \text{if } x \geq 0, \\ \infty & \quad \text{otherwise} \end{cases} $$ is closed and convex, and $\partial f(0) = \emptyset$ despite the fact that $f(0)$ is finite.

  • $\begingroup$ This is really strange. I thought that intuitively, a function only becomes non-subdifferentiable if the function goes to $\infty$ at some point at the boundary. Maybe I am stuck thinking of the x^2 over open domain and $+\infty$ else where. But now looking at the graph of -sqrt(x), it seems that the condition is the slope of the line becomes vertical. $\endgroup$ – Sanjay Gupta Sep 16 at 1:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.