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This is from The Way of Analysis by Strichartz.

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I believe see a neighborhood of $0$ such that $x_1 < x_2 \implies f(x_1) \le f(x_2)$.

Why is this zig zag function monotone increasing at $0$, but not monotone increasing in a neighborhood of $0$?

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    $\begingroup$ I have a couple of questions: what does it mean to be monotone increasing at a point? What is the definition of the function in question? $\endgroup$ – Theo Bendit Nov 5 '17 at 2:43
  • $\begingroup$ Presumably there are to be infinitely many wiggles, say hitting the $x$-axis at $1/n$ for every $n\ne0$. $\endgroup$ – Lubin Nov 5 '17 at 2:45
  • $\begingroup$ @TheoBendit I've included the defintions $\endgroup$ – Al Jebr Nov 5 '17 at 2:46
  • $\begingroup$ It's not increasing on any neighbourhood of $0$ because any such neighbourhood will contain a downwards "zag". (The picture could perhaps be better - I believe it's trying to communicate a "fractal" zig-zag as Lubin described.) $\endgroup$ – Anthony Carapetis Nov 5 '17 at 2:51
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I think the idea is that the function is fractal, in the sense that as we approach $0$ from either side, the function has smaller spikes with increasing frequency. Indeed, we can see that the function $f$ is such that $f\le 0$ for $x<0$, $f\ge 0$ for $x>0$, and $f=0$ at $x=0$.

$f$ is monotone at $0$, because given $x_1<x_0<x_2$ we have that $f(x_1)\le f(x_0)=0$ and $f(x_2)\ge f(x_0)=0$ so that $f(x_1)\le f(x_0)\le f(x_2)$. On the other hand, because of this spiking behavior, any neighborhood of $0$, $N=\{x\in \mathbf{R}: \lvert x\rvert<\epsilon\}$, contains an entire spike, which is clearly not a monotone function, since it increases, and then decreases. So $f$ is not monotone in a neighborhood of $0$.

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