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Please someone tell me why $\max\{0,x\}$ is not a function at $x=0$. I always learned that to fail the vertical line test the function's graph should have different values for the same input. However, this function apparently has those different values equal to each other. Not coincidentally, the book shows a line there and says $max\{0,x\}$ is not a function at the point. I am really asking for help, not just being rhetorical.

tropical hyperplane $\max{0,x}

David Scholz
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Ohio skateboarder . 7
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    Where is $\text{max}{0,x}$ in your photo of textbook? I can't see it anywhere. The function $\text{max}{0,x}$ as is usually understood, takes value $0$ for $x\leq 0$ and $x$ for $x\geq 0$. These coincide (as you point out) at $x=0$. So it is defined and has value $0$ at $x=0$ – user829347 Feb 13 '22 at 21:27
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    What does the plus sign in the circle and “dartboard” symbol mean next to the arrow?. It seems like $\max(0,x=0)=\max(0,0)=0$. – Тyma Gaidash Feb 13 '22 at 21:34
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    In case $\max{x,0}$ is not a function, maybe you however can agree that $\frac{1}{2}(x+|x|)$ is a function ? – Maksim Feb 13 '22 at 21:43
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    if $x = y$, $\max (x, y) = x = y$. – Essaidi Feb 13 '22 at 21:45
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    In tropical geometry, $x\oplus y:=\min{x,y}$ not $\max{x,y}$. So, $0\oplus(0\odot x_1)$ reads $\min{0,x_1}$. Moreover, it is nowhere said in this book that this is not a function. – KBS Feb 13 '22 at 21:49
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    @KBS I know essentially nothing about tropical geometry, but when I checked the Wikipedia article on it, they say that depending on the convention one uses $\max$ or $\min$ to define $\oplus$, and that the two semirings obtained are isomorphic. – projectilemotion Feb 13 '22 at 22:09
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    @projectilemotion In the book the OP is using, it is clearly defined as min. For information, the book is "Tropical Geometry and Mirror Symmetry" by Mark Gross. – KBS Feb 13 '22 at 22:11
  • So the line is not a tropical hyperplane after all? Apparently that is my mistake, I actually was reading a richard stanley hyperplane arrangement book so the figure kind of looked like one to me at the time... – Ohio skateboarder . 7 Feb 14 '22 at 02:18
  • max(0,x) is a function , also for $x=0$. The title does not make sense. – Peter Feb 14 '22 at 20:24

1 Answers1

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Well by definition

$$\max\{0,x\}=\begin{cases} x, & x>0, \\ 0, & x\leq 0 \end{cases},$$

which is indeed a function $\mathbb{R}\to\mathbb{R}$. Your picture does not show this function, however, so I don't know how you made that connection.

Lorago
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  • If it's a function like you say, then it doesn't yield a tropical hyperplane right? That's all I had time for studying math currently it's a bit hard to interpret all this new stuff in succession .. – Ohio skateboarder . 7 Feb 14 '22 at 02:30