gnometorule
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 Jan 4 awarded Nice Answer Dec 17 awarded Yearling May 29 comment Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. @marco11: converges from below - $x_i\le x_{i+1}$, and $x_i \rightarrow c$. May 29 comment Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. But then (using monotonicity) we can sum those differences, and would get that the function explodes to $\infty$, which it can't (bounded by some $m$). Maybe this helps? :) May 29 comment Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. The idea is: if along a sequence converging from the right to $c$ the corresponding function values converge, you are done by monotonicity (all other sequences are sandwiched by this one -monotonicity). Assume that doesn't happen: then no matter how close you get to $c$, the difference of two function values must not get smaller than some fixed $\epsilon$ (or it converges as it would be a Cauchy sequence). But then pick 2 more such values even closer to $c$...and the same applies again: their difference must be larger than $\epsilon$, etc. May 29 comment question about direct sum of vector fields and preservation under quotient spaces For posterity, the discussion was about an old counter example which didn't quite work. May 29 comment question about direct sum of vector fields and preservation under quotient spaces the new counter should work. May 29 revised question about direct sum of vector fields and preservation under quotient spaces added 10 characters in body May 29 comment question about direct sum of vector fields and preservation under quotient spaces I meant the first line as before, but second line as $V= \pi^{-1}(\pi(V_1) \oplus \pi^{-1}\pi(V_2)$. I assume what you changed is fine too, but I like this (old first line, plus this second line) formulation better as - to me - it's more clear what's going on. :) May 29 revised question about direct sum of vector fields and preservation under quotient spaces added 474 characters in body May 29 comment question about direct sum of vector fields and preservation under quotient spaces There should be a typo in 2): I think you have $\pi^{-1}(\pi(V_1)$ etc on the RHS (or the function is not well-defined). May 29 revised question about direct sum of vector fields and preservation under quotient spaces deleted 20 characters in body May 29 answered question about direct sum of vector fields and preservation under quotient spaces May 29 comment Solve Inequality for $|x|$ You essentially derived an inequality B from an inequality A. All x solving B will solve A, but you haven't shown that those are all solutions. For the general case, solve the inequality by looking at $x$ in $[2, \infty)$, $(-3, 2)$, and $(-\infty, -3)$ (inequality is not defined at $-3$). May 28 revised Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. edited body May 28 revised Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. edited body May 28 revised Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. edited body May 28 revised Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. edited body May 28 revised Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist. added 43 characters in body May 28 answered Let $f:(a,b)\rightarrow \mathbb{R}$ be non-decreasing i.e. $f(x_1)\leq f(x_2)$ and let $c \in (a,b)$. Show that limits from both sides exist.