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# An interesting everywhere-differentiable function

A graph has equation $f(x)=-\frac{1}{n}x-\frac{1}{n}\sin x+\left|\frac{1}{n}\sin x-\frac{1}{n}x\right|$

for a non-zero integer $$n$$. Let's look at the graph for $$n=1$$:

Imgur

Hey look, no rough non-differentiable edges. Cool! Try other values of $$n$$ to see what happens.

Can you create any other interesting graphs with the absolute value function that are differentiable everywhere?

Note by Daniel Liu
2 years, 7 months ago

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Sure, replace $$n$$ with $${ n }^{ 2 }$$ Another one, which is technically not a function, is $$\pm \sqrt { { x }^{ 2 }-\left| { x }^{ 3 } \right| }$$, looks kind of like a lemniscate, is differentiable everywhere except at $$x=-1, x=1$$, and has an area of $$16/15$$. But the use of $$\pm$$ makes this claim of differentiability at $$x=0$$ iffy at best. · 2 years, 7 months ago

well, $$n$$ is a dummy variable, so changing it doesn't really affect anything. If you meant $$x$$, then the function isn't very clever nor differentiable everywhere; it just looks like a regular sine function with a little nick in the middle.

The second one is pretty cool. One can manage to graph the whole thing at once on desmos by using $|y|=\sqrt{x^2-|x^3|}$

Here is a comparison of the graph to the lemniscate: https://www.desmos.com/calculator/vfw1amglgn · 2 years, 7 months ago

Yes, one can see that tiny difference in areas, the area of the lemniscate being exactly $$1$$. As for the other, replace $$n$$ with $$1+{ x }^{ 2 }$$. · 2 years, 7 months ago

Ooh, that graph looks pretty interesting. It might be convincing enough to pass as one non-piecewise graph, although people would have a hard time finding the equation for it. · 2 years, 7 months ago

What program did you use to draw the graph? Thanks for posting this. I will post something related to graphs too. · 2 years, 7 months ago