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$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$

Like half-circles connected to each other to look like waves:

plot

Its plot looks smooth, but the function is actually not smooth.

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This is like proving that pink unicorns do not exist, of course... but my guess is that your function does not have a standard name. –  Mariano Suárez-Alvarez Apr 28 '11 at 13:46
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Henceforth it shall be called the Vio periodic function! –  Raskolnikov Apr 28 '11 at 16:30
    
I've seen the version where all the semicircles are above the horizontal axis... this one though, maybe you get dibs on naming it. :) –  J. M. Apr 28 '11 at 17:22
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Looks like "poor man's sine that can be drawn with a compass". –  Vi0 Apr 28 '11 at 18:23
    
@Raskolnikov, (actual nick is "_Vi", but is's not StackExchange-compliant) –  Vi0 Apr 28 '11 at 18:24
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