# A natural function with discontinuities (Part 1)

The following tidbit that was published on the American Mathematical Monthly’s Facebook page caught my attention:

Here’s the relationship between $r$, $R$, and $\theta$ in case it isn’t clear from the description. The gray sector is determined by $r$ and $\theta$, and then the blue circle with radius $r$ is chosen to enclose the sector.

Unfortunately, there was typo for the third case; it should have been $r = R \sin \frac{1}{2} \theta$ if $90^\circ \le \theta \le 180^\circ$. Here’s the graph if $R = 1$, using radians instead of degrees:

As indicated in the article, there’s a discontinuity at $t=0$. However, the rest of the graph looks nice and smooth.

Here’s the graph of the first derivative:

The first derivative is continuous (and so the original graph is smooth). However, there are obvious corners in the graph of the first derivative, which betray discontinuities in the graph of the second derivative:

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