Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Why do you need notation? Just use English. – Qiaochu Yuan Dec 14 '12 at 10:30
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...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.