Saw-tooth function

Quite sometime ago, I noticed something interesting regarding the saw-tooth function.

\[y=\begin{cases} \lim _{ n\rightarrow \infty }{ f^{ n }\left( x \right) -x } \quad \text{ n is an even integer} \\ \lim _{ n\rightarrow \infty }{- f^{ n }\left( x \right) -x } \quad \text{ n is an odd integer} \end{cases}\] \[f\left( x \right) =\sin \left( x \right) -x\] appears to describe the saw-tooth function.

I have no idea why this is so. If anybody knows, feel free to comment it. Thanks.

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Comments

Your limit is confusing to read. I'm guessing that what you want is

$y = \lim_{n \rightarrow \infty} ( -1)^n f^n (x) - x.$

Calvin Lin Staff - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$