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Consider $f : U \subset X \to \mathbb{R} \cup \{-\infty, +\infty\}$. $X$ is assumed to be a Banach space (or $\mathbb{R}^n$) and $U$ is an open subset.

Any suggestions about references treating differentiability of such a function that is allowed to take infinite values.

Thank you.

user2015
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  • Do you ask for an useful definition or do you know a definition and want a reference to it? – user251257 Feb 16 '16 at 00:50
  • I ask for a course where the author does not restrict him self to the common framework of real (finite) functions.

    In optimization, we allow the value of the function to be $-\infty$ and $+\infty$.

     – user2015 Feb 16 '16 at 07:40
  • In context of convex analysis/optimization there is the concept of subgradient. There is also the concept of metric derivative, if you define a metric on the extended real line. – user251257 Feb 16 '16 at 07:46
  • First of all thank you user251257 for your interest. – user2015 Feb 16 '16 at 17:24
  • Do you mean that we define differentiability just at $a$ such that $f(a)$ is finite to go back to the classical definition with real valued functions ? – user2015 Feb 16 '16 at 17:33
  • for subgradient $f(a)$ needs to be finite. But it is not the classical derivative. You could also consider weak derivative in $L^p$ sense or distribution sense. But they are not defined point wise. Since you mentioned optimization, subgradient is most likely the object you need. But just ask your teacher for reference. – user251257 Feb 16 '16 at 18:54

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