I am reading a paper where the sign function is defined as the following $$\text{sign}(u)=\begin{cases}1, & u>0 \\ [-1,1], & u=0 \\ -1, & u<0 \end{cases}.$$ And the equations with equal sign is replaced by the belong sign $\in$, for example, $p$ and $S$ are two functions with the relationship $$p\in P(S)+\text{sign}\left(\frac{\partial S}{\partial t}\right).$$ I am so confused about the meaning when the value of a function is a set.
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1I think the author of the paper is saying that $sign (u)$ can be taken to be any number between $-1$ and $+1$ when $u=0$. – geetha290krm Nov 22 '22 at 06:12
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May be it's family of functions. – zkutch Nov 22 '22 at 06:21
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It’s a set-valued map. – Theo Bendit Nov 22 '22 at 07:07
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Maybe this should be $\text{sign}(u)=\begin{cases}{1}, & u>0 \ [-1,1], & u=0 \ {-1}, & u<0 \end{cases}$ ? However, what I'm giving is not the usual "sign" function in which the range (i.e. set of outputs) is the $3$-element set ${-1,0,0}.$ Here the range is the $3$-element set ${{-1},,[0,1],,{1}}.$ I don't know if this is what the author intends. The equation you've given for "sign" seems a bit strange. The author is also possibly abusing notation, but for something unusual like this, an author should not be doing this, at least not without an explanatory comment somewhere. – Dave L. Renfro Nov 22 '22 at 09:15
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1By the way, for questions like this you should provide the title, author(s), and year for the paper, as well as on which page the equation appears (and if the page is full of text and/or equations, specify somewhat where in that page). And if the paper is online, a link to the paper. The reason for specifying the paper is that notation varies sometimes across various areas of math, and across time periods. Knowing what paper you're asking about can go a long ways toward answering your question. – Dave L. Renfro Nov 22 '22 at 09:20