$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$
Like half-circles connected to each other to look like waves:
plot http://vi-server.org/vi/_/func.png
Its plot looks smooth, but the function is actually not smooth.
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5This 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-Álvarez Apr 28 '11 at 13:46
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1Henceforth it shall be called the Vio periodic function! – Raskolnikov Apr 28 '11 at 16:30
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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. ain't a mathematician Apr 28 '11 at 17:22
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1Looks like "poor man's sine that can be drawn with a compass". – Vi0 Apr 28 '11 at 18:23
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@Raskolnikov, (actual nick is "_Vi", but is's not StackExchange-compliant) – Vi0 Apr 28 '11 at 18:24
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@Mariano Suárez-Alvarez, But there can related named function (e.g. negated, with other period). – Vi0 Apr 28 '11 at 18:26
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I don't know if it has a name, but it is a very ugly thing. – JT_NL Apr 28 '11 at 20:46
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@Jonas T, Why ugly? It's even continuous. And circle is not ugly, so that semi-circle is unlikely to be ugly, and this is the most logical way to make a "semicircle function". – Vi0 Apr 28 '11 at 21:10