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Supposing a function can assume any value under a curve. As an example, we have a curve f(x). However, my actual function is that for a given value of x, the value y can be anything lower than f(x).

How would I write down this multi-valued function? And is calculating derivative of this function possible?

zynga
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  • You mean the value can be in a range? That's not a function – Sean Roberson Sep 25 '18 at 08:07
  • yes. That is not a function in the true sense. But can it be written down, say as a complex function? – zynga Sep 25 '18 at 08:09
  • Do You mean something like $\Phi:D\rightarrow {[0,r]:r\in\mathbb{R}^{\geq 0}},x\mapsto [0,|f(x)|]$ where $D\subseteq\mathbb{R}$ is the domain of the function $f$? For a derivative You would need at least some kind of metric on the codomain of this function $\Phi$ though. – Peter Melech Sep 25 '18 at 10:21
  • Cool! What kind of metric though? – zynga Sep 25 '18 at 17:32

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