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.