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I want to define a good notation for the number of how often the function $f:\mathbb{R}\rightarrow \mathbb{R}$ changes its sign in the interval $I\subset \mathbb{R}$ significantly.

By significantly I mean that the function changes its sign forgiven $\epsilon>0$ if there are values $x_1,x_2\in I$ with $f(x_1)>\epsilon$ and $f(x_2)<-\epsilon$.

The one notation I could think of was $$ \max \{n\in\mathbb{N} :\text{there are }x_1,\ldots, x_n\in I\text{ with }x_i<x_{i+1}, \operatorname{sgn}(1_{\mathbb{R}\setminus[-\epsilon,\epsilon]}(f(x_{i-1}))\neq \operatorname{sgn}(1_{\mathbb{R}\setminus[-\epsilon,\epsilon]}(f(x_{i}))\} $$

However, this feels notational clumsy for such an intuitive concept. Are there better (and more standard) notations for that number?

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Why do you need notation? Just use English. –  Qiaochu Yuan Dec 14 '12 at 10:30
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1 Answer 1

up vote 2 down vote accepted

Sometimes it takes some time to notice the obvious... I considered Qiaochu's comment used a formally clear definition in plain english:

Let $f\in C(\mathbb{R})$ and $\epsilon>0$.

  • The function $f$ features an $\epsilon$-significant sign change between $x,x'\in\mathbb{R}$, $x<x'$ if
    1. $|f(x)|=|f(x')|=\epsilon$,
    2. $f(x)=-f(x')$
    3. and $f(y)\in[-\epsilon, \epsilon]$ for all $y\in[x,x']$
  • The oscillation number $\omega_\epsilon(f, I)$ in the interval $I\subset\mathbb{R}$ is the number of $\epsilon$-significant sign changes of $f$ within the interval $I$.

Sometimes, the plain and pure solutions are just to obvious to be considered...

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