Here’s the relationship between , , and in case it isn’t clear from the description. The gray sector is determined by and , and then the blue circle with radius is chosen to enclose the sector.
Unfortunately, there was typo for the third case; it should have been if . Here’s the graph if , using radians instead of degrees:
As indicated in the article, there’s a discontinuity at . 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:
I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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One thought on “A natural function with discontinuities (Part 1)”