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 \(n=1\):

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?

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TopNewestSure, 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. – Michael Mendrin · 2 years, 7 months ago

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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 – Daniel Liu · 2 years, 7 months ago

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– Michael Mendrin · 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 \(1+{ x }^{ 2 }\).Log in to reply

– Daniel Liu · 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.Log in to reply

– Adrian Neacșu · 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.Log in to reply

modulus functions !! they're awesome pieces very useful for manipulation ................ nice one !!............. seems like the graph changed it's mind after entering the positive x-axis!! :-p – Abhinav Raichur · 2 years, 7 months ago

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