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Hometriangular-wave function

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# triangular-wave function

The arcsine is the inverse function of the sine. Therefore the composition function

$f:\,x\mapsto\arcsin(\sin{x})$ |

is the identity map $x\mapsto x$ on the interval $[-\frac{\pi}{2},\,\frac{\pi}{2}]$. On this interval, the inner function $\sin$ increases monotonically and continuously from its least value $-1$ to its greatest value 1; then the outer function $\arcsin$ (i.e. the angle corresponding the sine value) and the whole composition correspondingly grows from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. On the next equally long interval $[\frac{\pi}{2},\,\frac{3\pi}{2}]$, when the inner function decreases from 1 to $-1$, the composition thus decreases from $\frac{\pi}{2}$ to $-\frac{\pi}{2}$, evidently again linearly. We have now run through a period interval $[-\frac{\pi}{2},\,\frac{3\pi}{2}]$ of the inner function and the composition $f$ and obtained a wedge-formed portion ($\wedge$) of the graph. Because of the periodicity, the whole graph of $f$ consists of such successive wedges and thus looks like a saw blade. The triangular-wave function is continuous. Its derivative (away from the singular points $\frac{\pi}{2}+n\pi,\,n\in\mathbb{Z}$) is a square-wave function.

Sometimes, such a function is called a saw-tooth function, although this name usually refers to a discontinuous function with graph consisting of either ascending ($/$) or descending ($\backslash$) line segments with jumps.

## Mathematics Subject Classification

53A04*no label found*26A06

*no label found*

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