envelope of a function
Consider $f:\mathbb{R}\to \mathbb{R}$ a real function of real variable.
We call the upper envelope of $f$ to the function^{}:
$$
similarly the lower envelope of $f$ is the function:
$$
The envelopes have the following properties: (in this list ${\mathrm{env}}_{\ast}$ represents either the upper or lower envelope)

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${\mathrm{env}}_{inf}(f)(x)\le f(x)\le {\mathrm{env}}_{sup}(f)(x)$

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${\mathrm{env}}_{sup}(f)={\mathrm{env}}_{inf}(f)\iff f\text{is continuous}$

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${\mathrm{env}}_{sup}(f)(x){\mathrm{env}}_{inf}(f)(x)=\text{oscillation of}f\text{at}x$

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${\mathrm{env}}_{inf}f={\mathrm{env}}_{sup}(f)$
Title  envelope of a function 

Canonical name  EnvelopeOfAFunction 
Date of creation  20130322 15:44:22 
Last modified on  20130322 15:44:22 
Owner  cvalente (11260) 
Last modified by  cvalente (11260) 
Numerical id  5 
Author  cvalente (11260) 
Entry type  Definition 
Classification  msc 26A99 