User contributions for Ccatrett
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2 April 2023
- 19:2619:26, 2 April 2023 diff hist 0 Graph of Continuous Mapping to Hausdorff Space is Closed in Product/Proof 1 No edit summary current
21 June 2022
- 04:4104:41, 21 June 2022 diff hist +1,899 Graph of Continuous Mapping to Hausdorff Space is Closed in Product No edit summary Tag: Visual edit: Switched
13 June 2022
- 08:4208:42, 13 June 2022 diff hist +10 Continuous Midpoint-Concave Function is Concave No edit summary
- 08:4108:41, 13 June 2022 diff hist +198 Continuous Midpoint-Concave Function is Concave No edit summary
- 08:3408:34, 13 June 2022 diff hist +1,046 Continuous Midpoint-Convex Function is Convex No edit summary
- 08:0308:03, 13 June 2022 diff hist +1 Midpoint-Convex Function is Rational Convex No edit summary
- 08:0308:03, 13 June 2022 diff hist +17 Midpoint-Convex Function is Rational Convex No edit summary Tag: Visual edit: Switched
- 08:0208:02, 13 June 2022 diff hist +68 Midpoint-Convex Function is Rational Convex links to rational convex definition and forward backward induction are provided for supporting the underlying logic of this proof Tag: Visual edit
- 07:5907:59, 13 June 2022 diff hist +305 Midpoint-Convex Function is Rational Convex No edit summary Tag: Visual edit: Switched
- 07:4707:47, 13 June 2022 diff hist +2,350 N Midpoint-Convex Function is Rational Convex Created page with "= Theorem = Let $f:I\to\mathbb{R}$ be a midpoint convex function defined on a real, non-empty interval $I$. Then $f$ is rational convex. == Proof == It suffices to show that for each $n\in\mathbb{N}$ and for any choice of $n$ elements $x_1,\dots,x_n$ in $I$, we have that $$f\left(\frac{x_1+\dots+x_n}{n}\right)\leq\frac{f(x_1)+\dots+f(x_n)}{n}$$ via forward-backward induction. The statement holds for $n=0$ vacuously and $n=1$ as $f(x/1)=f(x)/1$ for each $x\in I$. If..." Tag: Visual edit: Switched
- 06:3106:31, 13 June 2022 diff hist −3 Definition:Rational Convex No edit summary Tag: Visual edit: Switched
- 06:3006:30, 13 June 2022 diff hist +218 N Definition:Rational Convex Created page with "== Definition == Let $f$ be a real function defined on a real interval $I$. $f$ is '''rational convex''' on $I$ if and only if: $$\forall x,y\in I\,\forall t\in[0,1]\cap\mathbb{Q}:f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$"
- 05:3505:35, 13 June 2022 diff hist +53 Talk:Continuous Image of Compact Space is Compact/Corollary 3 No edit summary current
- 05:3405:34, 13 June 2022 diff hist +1,029 N Talk:Continuous Image of Compact Space is Compact/Corollary 3 Created page with "Since $f[S]$ compact, then $f[S]$ is closed, hence $f[S]=\overline{f[S]}$. Let $U\ni x$ be an open neighborhood of $\alpha\,\colon=\sup f[S]$. Then there is an $r>0$ such that $B_r(\alpha)=(\alpha-r,\alpha+r)\subseteq U$. From Characterizing Property of Supremum of Subset of Real Numbers[1], there is a $y\in f[S]$ such that $\alpha-r<y\leq\alpha$, thus $\alpha$ is an adherent point of $f[S]$. As a point is adherent to $f[S]$ if and only if it is in $\overline{f[S]}$, the..."