All public logs

Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).

• 07:47, 13 June 2022 Ccatrett talk contribs created page 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
• 06:30, 13 June 2022 Ccatrett talk contribs created page 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: $$\forall x,y\in I\,\forall t\in[0,1]\cap\mathbb{Q}:f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$")
• 05:34, 13 June 2022 Ccatrett talk contribs created page 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...")