Timeline for Goldowsky-Tonelli theorem for upper semi continuous function
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 11, 2019 at 11:58 | vote | accept | Adam | ||
| Oct 13, 2019 at 18:09 | |||||
| Oct 11, 2019 at 11:07 | comment | added | Pietro Majer | This is elementary. Recall $f:(a,b)\to\mathbb{R}$ is convex iff the incremental ratio ${f(y)-f(x)\over y-x}$ is increasing in both variables. As a consequence, at any point $x$, $f$ has both right derivative, $f_+'(x)=\inf_{y>x}{f(y)-f(x)\over y-x}$ and left derivative $f_-'(x)=\sup_{z<x}{f(z)-f(x)\over z-x}$, and for all $x<y$ in $(a,b)$ $$f_-'(x)\le f_+'(x) \le {f(y)-f(x)\over y-x} \le f_-'(y)\le f_+'(y).$$ | |
| Oct 11, 2019 at 10:12 | comment | added | Adam | Why for $0<x<y$ we have $f^{'}(x)\leq \frac{f(y)-f(x)}{y-x}\leq f^{'}(y)?$ | |
| Oct 11, 2019 at 7:15 | comment | added | Pietro Majer | Yes, what exactly is not clear? | |
| Oct 10, 2019 at 21:20 | comment | added | Adam | Did you show that it is decreasing? Could you please explain it a bit more? | |
| Oct 10, 2019 at 19:25 | history | answered | Pietro Majer | CC BY-SA 4.0 |