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What types of infinitely differentibale functions $f:\mathbb{R} \rightarrow \mathbb{R}$ have the following property?

$$\lbrace x : |f(x)| = 1 \rbrace \nsubseteq cl \lbrace x : |f(x)| > 1 \rbrace$$

where $cl$ denotes closure of a set.

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Why does the title mention infinitely differentiable functions? This seems to be unrelated to your question. –  Antonio Vargas Jan 20 '13 at 3:43
    
I forgot to put it in the description, I'm interested in infinitely differentiable functions having the said property. –  user58191 Jan 20 '13 at 3:47
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Every infinitely differentiable function such that $|f(x)|=1$ at least once and $|f(x)|\leq 1$ for all $x$ is an example (this condition is sufficient but not necessary). $f(x)\equiv 1$ is the simplest. What do you mean by "what types"? Do you just want some examples? –  Jonas Meyer Jan 20 '13 at 3:51
    
@JonasMeyer: I have some examples at hand, but am interested if there are some pathological examples –  user58191 Jan 20 '13 at 4:00
    
What are your examples, and what do you mean by pathological? –  Jonas Meyer Jan 20 '13 at 4:02

1 Answer 1

A continuous function has this property if and only if there is a point $x$ for which $|f(x)| = 1$ and $|f(y)| \leq 1$ for all $y$ in a neighborhood of $x$.

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In the infinitely differentiable case, necessary conditions for such $x$ include $f'(x)=0$. If $f(x)=1$, then $f''(x)\leq 0$, and if $f(x)=-1$, then $f''(x)\geq 0$. –  Jonas Meyer Jan 20 '13 at 4:00

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