Find a Wave Function with a Chaotic, Unbounded, Non-monotonic Amplitude and Wavelength What family of sinusoidal functions has chaotic, unbounded, non-monotonic amplitudes and wavelengths where neither become neither infinitesimal nor infinite as x approaches positive infinity?
 A: On what domain? $x^3-x$ is neither monotonic nor bounded on $\mathbb R$.
How about $f(x)=0$ for all $x$ irrational and for $x=0$, and $f(x)= \dfrac 1x$ otherwise? Not sinusoidal, average zero (zero almost everywhere), not monotonic over any interval?
A: What about the function $f(x) = x \cdot \cos(x)$? Here is its graph (from Wolfram Alpha):


The function $f(x)=x\cdot\left[\sin(x)+\sin(2x)\right]\cdot\cos(2x)$ seems to fit your criteria.
Here is a graph:

If we graph $\vert f(x) \vert$, we can see that the absolute values of the extrema are not monotonic:

A: How about 
$
\big(x + \sin x \big) \cos (\pi x)
$?
It seems to satisfy all the conditions you've given so far.


Amendment:
You can play this game all you want, and I'm going to stop soon, but here's a result that seems to be everything you're hoping for. It combines my previous answer and @TreeHouse196's. I'm using mathematica/wolfram alpha for the plots, and I recommend you do too. Wolfram Alpha is a popular free online tool.

A: I think $f(n)={(-1)}^{n}n$, over non negative integers is good example of an unbounded function which is not monotonic.
sorry sir,it was a typo...
