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The plot of function & the function definition is here: https://www.desmos.com/calculator/p4y97ww37l

Edit In case the above link isn't working,here's the function definition:

$$y=\left \lceil x \right \rceil.sin\frac{\pi}{\left \lceil x+1 \right \rceil}$$

Spikes are also observed at x=(2^n)-1 where n=1,2,3...

Can anyone provide any explanation for the observed peaks?

Also what would be the Domain & Range of this function? Please also provide a method for finding it without plotting it.

Mark McClure
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kryptoknight
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  • Cool...it seems to converge to $\pi$ as $x\to\infty$. – Adriano Aug 05 '13 at 19:41
  • Same is the case for x $\rightarrow -\infty$ – kryptoknight Aug 06 '13 at 11:12
  • The page doesn't load for me. Could you edit your post to include the definition? – Antonio Vargas Aug 07 '13 at 06:01
  • Here's the function:

    $$y=\left \lceil x \right \rceil.sin\frac{\pi}{\left \lceil x+1 \right \rceil}$$

     – kryptoknight Aug 07 '13 at 13:44
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    The spikes are an artifact of the plotting system, and are not "real". Observe that the expressions $\lceil x \rceil$ and $\lceil x + 1 \rceil$ have identical step discontinuities. Examining the continuous version $x \sin \frac {\pi}{x + 1}$ shows what happens when $x$ is large, and the behaviour near but outside the interval $(-2, -1]$ is dependent upon a few fixed values. The value when $x \in (-2, -1]$ is undefined. – Steve Powell Aug 07 '13 at 15:51

1 Answers1

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Clearly, the only reason the function would be undefined would be a division by zero, i.e. as long as $x$ is not in $[-1,0)$, the function is defined. Thus, the domain is $$\{x:x<-1 \text{ or } x\geq 0\}.$$

Now, the function is constant on any interval of the form $[n,n+1)$, where $n\in\mathbb Z$ (although, again undefined when $n=-1$). Thus, one way to write the range is as

$$\{n\sin(\pi/(n+1)): n\in\mathbb Z \setminus \{-1\}\}.$$

I don't know that there is any easier way to describe the set, although we can certainly see some of it's characteristics. Using the fact that $$\lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1,$$ it's not too hard to see that $$\lim_{n\rightarrow \pm\infty} n\sin\left(\frac{\pi}{n+1}\right) = \pi.$$ Thus, $\pi$ is a cluster point of the range, while all other points are isolated. Beyond that, I think a graph is a reasonable thing to have. In the following picture, we see the graph together with a portion of the range plotted as red dots on the $y$-axis.

enter image description here

Mark McClure
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