What is the equation for this wave-like curve whose amplitude decreases to $0$ and frequency increases to infinity as $x\to 0$? Imagine a curve like a sine wave that is mutated thus: For an increasing $X > 0$, and decreasing $X < 0$ its frequency decreases by the same rate that its amplitude increases. Therefore, as $X$ approaches $0$ (from either direction) its frequency increases infinitely and its amplitude decreases infinitely by the same rate. Can you help me? $x\sin(1/x)$ was suggested and it looks really perfect for $x < 0.25:$ ![enter image description here]( However for $x > 1$ it very quickly fails to continue on with the same pattern: ![enter image description here]( Just to be clear, as I imagine what this curve looks like I see a curve that looks the same no matter if you're zoomed in close to 0 or zoomed out. If you're origin is in the center the curve will look the same on any scale. This is because the amplitude and frequency are changing at the same rate.

One such function would be $f(x) = x \cdot \sin(\pi\log_2(x))$, which has zeros at $x = 2^n$, $n\in\mathbb{Z}$ and between two consecutive zeros $2^n$, $2^{n+1}$ reaches an amplitude between $2^n$ and $2^{n+1}$.

This has the right scaling behaviour: "Zooming in" by a factor of $2$ on the graph of the function corresponds to looking at the graph of $2f(\frac{x}{2}) = 2\frac{x}{2} \sin\left(\pi (\log_2(x)-\log_2(2))\right) = x \sin(\pi \log_2(x)) = f(x)$. Hence, the graph does not change if you zoom in or out by a factor of $2$.

Here's the graph of $|x|\cdot\sin(\pi\log_{1.618}(|x|))$ (which is invariant under scaling by a factor of $1.618$ instead of $2$), courtesy of Legit Stack:

![Graph of the function](