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:

$\operatorname{env}_{\sup}(f)(x)=\inf_{\epsilon}\{\sup\{f(y):\epsilon>0,\>|y-x|% <\epsilon\}\}$

similarly the lower envelope of $f$ is the function:

$\operatorname{env}_{\inf}(f)(x)=\sup_{\epsilon}\{\inf\{f(y):\epsilon>0,\>|y-x|% <\epsilon\}\}$

The envelopes have the following properties: (in this list $\operatorname{env}_{\ast}$ represents either the upper or lower envelope)

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$\operatorname{env}_{\inf}(f)(x)\leq f(x)\leq\operatorname{env}_{\sup}(f)(x)$
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$\operatorname{env}_{\sup}(f)=\operatorname{env}_{\inf}(f)\iff f\text{is continuous}$
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$\operatorname{env}_{\sup}(f)(x)-\operatorname{env}_{\inf}(f)(x)=\text{% oscillation of}f\text{at }x$
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$\operatorname{env}_{\inf}f=-\operatorname{env}_{\sup}(-f)$