In the comment to this answer: http://math.stackexchange.com/a/144819/21919

the last display formula shows $...-f(x\,0)$ instead of $\ldots-f(x_0)$, though it is not a misprint. I tried to edit it, and to post the whole comment anew, but this little bug persists. It shows in Chrome (version 27.0.1453.116 m) and IE 8 under Windows 7.

Here's my original TeX code:

Nice! However, I had hard time understanding the last sentence, so maybe it is worth to supply a few more details as follows: Let $V$ denote the variation over the partition chosen within $\epsilon/2$ of $TV(f_{[x_1,x_0]})$ as described above. Then $$TV(f_{[x_1,x_0]})< V+\epsilon/2.$$ Also, $V-|f(x)-f(x_0)|$ is some variation over interval $[x_1,x]$, so we have: $$TV(f_{[x_1,x]})\ge V-|f(x)-f(x_0)|.$$ Finally, $TV(f_{[x_1,x_0]})=TV(f_{[x_1,x]})+TV(f_{[x,x_0]})$, so that: $$TV(f_{[x,x_0]})=TV(f_{[x_1,x_0]})-TV(f_{[x_1,x]})<(V+\epsilon/2)-(V-|f(x)-f(x_0)|)<\epsilon/2+\epsilon/2=\epsilon.$$

In the comment to this answer: https://math.stackexchange.com/a/144819/21919

the last display formula shows $...-f(x\,0)$ instead of $\ldots-f(x_0)$, though it is not a misprint. I tried to edit it, and to post the whole comment anew, but this little bug persists. It shows in Chrome (version 27.0.1453.116 m) and IE 8 under Windows 7.

Here's my original TeX code:

Nice! However, I had hard time understanding the last sentence, so maybe it is worth to supply a few more details as follows: Let $V$ denote the variation over the partition chosen within $\epsilon/2$ of $TV(f_{[x_1,x_0]})$ as described above. Then $$TV(f_{[x_1,x_0]})< V+\epsilon/2.$$ Also, $V-|f(x)-f(x_0)|$ is some variation over interval $[x_1,x]$, so we have: $$TV(f_{[x_1,x]})\ge V-|f(x)-f(x_0)|.$$ Finally, $TV(f_{[x_1,x_0]})=TV(f_{[x_1,x]})+TV(f_{[x,x_0]})$, so that: $$TV(f_{[x,x_0]})=TV(f_{[x_1,x_0]})-TV(f_{[x_1,x]})<(V+\epsilon/2)-(V-|f(x)-f(x_0)|)<\epsilon/2+\epsilon/2=\epsilon.$$