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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
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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
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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
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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.
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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.
May
26
revised demonstrate that $v_3 \perp (v_1-v_2)$
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May
26
revised for which values of the pair of integers $(n,k)$ is $p(n,k) =1+\frac{2^{k}-1}n$ is prime?
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