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I have a question regarding this graph, for $f(x)=x-[x]-\dfrac{1}{2}$, where $[\cdot ]$ denotes the greatest integer function. My question is about the graph, why does the first slop has an open circle at $(-3,-0.5)$? shouldn't it be a close circle since $f(-3)=-3-[-3]-\dfrac{1}{2}=-0.5?$

enter image description here

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    $\begingroup$ Yes, that's a bug in your plotting software. $\endgroup$ – enedil Mar 15 at 14:04
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    $\begingroup$ I agree. If $n$ is an integer, then $f(n)=-1/2$, not $0$. $\endgroup$ – TonyK Mar 15 at 14:05
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    $\begingroup$ I remember a similar problem you had here: math.stackexchange.com/questions/3113840/… you should finally quit using the site you mentioned there as the graphs there are erroneous. $\endgroup$ – trancelocation Mar 15 at 14:16
  • $\begingroup$ @trancelocation Yes, I remember your comment there, it was very helpful, and you are correct, many of the graphs there were wrong. Definitely will stop using it. $\endgroup$ – Friedrich Mar 15 at 14:23
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It looks as if your software may be treating $[n]=n-\frac12$ for integer $n$

That would be halfway between $\lim\limits_{x \to n^{-}} [x]=n-1$ and $\lim\limits_{x \to n^{+}} [x]=n$, but it is not a common usage of a floor function aimed at producing integers

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