I would like to know a single (not piecewise) continuous approximation of $x \bmod 1$ which gets sharper the more you increase a constant $c$. I do not want a series like Fourier, but I do want something that uses continuous trigonometric functions (though $\arcsin$ is fine).
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2$\arctan(\sin x/\cos x)$ has the shape you want. Now replace the $1/\cos x$ factor with a continuous approximation, say $\cos x/(c + \cos^2 x)$ as $c\to0$. – Nov 25 '19 at 02:35
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https://math.stackexchange.com/questions/2491494/does-there-exist-a-smooth-approximation-of-x-bmod-y – fGDu94 Nov 25 '19 at 03:00
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I've added an answer there now, FYI. – Nov 25 '19 at 11:06
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3@Rahul I think you meant $\arctan(\cos x/ \sin x)$ – MathPowers Nov 25 '19 at 16:16
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@Rahul Thank you! Can you put your comment as an answer? – ILoveMath2 Nov 25 '19 at 21:02
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I did. Didn't you see my last comment? – Nov 26 '19 at 02:35
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@Rahul I meant on this question. – ILoveMath2 Nov 27 '19 at 01:10
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Then no, I have already added an answer there that you can refer to. – Nov 27 '19 at 05:51