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I have a question regarding the stepwise function properties.

I have the stepwise function on the uniform grid:

\begin{equation}\label{step} f(t)=\begin{cases} 2, \ \ \ \ -2<t<-1 \\ 1, \ \ \ \ -1<t<0 \\ 2, \ \ \ \ 0<t<1 \\ 3, \ \ \ \ 1<t<2 \end{cases} \end{equation} which I plot on the re-scaled interval $[-\pi,\pi]$ is shown in Fig. 6.

In order to explain the indicator function, I add each intervals by the coordinates (circled in red).

They overlap, as you can see. Is this correct? Because I don't get the same as on wikipedias page:

https://en.wikipedia.org/wiki/Step_function

Thanks

enter image description here

Luthier415Hz
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    On the contrary, they do not overlap and instead your function as you have written it is undefined at those values. Note how in the linked wiki article they draw their graph with filled dots versus open dots to indicate which is the defined value for the particular values. To make your function defined for those circled values, you need to do something like replacing your $<$ signs with $\leq$ signs. – JMoravitz Apr 17 '23 at 12:10
  • So how would you draw the undefined values on that plot? Do you have an example , a link or something? – Luthier415Hz Apr 17 '23 at 12:43
  • @JMoravitz I re-did the calculations with $\le$ instead of $<$, they give the same plots and coefficients. – Luthier415Hz Apr 17 '23 at 12:49
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    Yes, but now half of those red circles are red dots instead of open circles. "How would you draw the undefined values on that plot": I would have emphasized they are undefined by drawing with open circles at those positions just like you have here already. "Do you have an example, a link or something" The link you already shared yourself. See also for more examples the search term "removable discontinuity" https://mathworld.wolfram.com/RemovableDiscontinuity.html – JMoravitz Apr 17 '23 at 12:52
  • @JMoravitz you mention half of the red circles should be red dots instead. Is the red dot included in the domain of one indicator function, and in the next, the same position should be a red circle, since it cannot be included at two same positions on the x-axis at the same time? – Luthier415Hz Apr 17 '23 at 14:11

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