envelope of a function

Consider $f:ℝ\to ℝ$ 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)

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

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

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

• ${\mathrm{env}}_{inf}f=-{\mathrm{env}}_{sup}(-f)$