# 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
7 years, 1 month 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.

- 7 years, 1 month 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

- 7 years, 1 month 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 $1+{ x }^{ 2 }$.

- 7 years, 1 month 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.

- 7 years, 1 month ago

What program did you use to draw the graph? Thanks for posting this. I will post something related to graphs too.

- 7 years, 1 month ago

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

- 7 years, 1 month ago