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 Jan 8 comment Is there a multiple function composition operator? Thank you for the answer. I'm not $\LaTeX$-ing this assignment (for reasons that are not relevant). It seems that no such notation exists, then? I was hoping someone could have come across something in a paper they read before. Jan 8 comment Is there a multiple function composition operator? Ah, I seem to have found a duplicate. Unfortunately, the answer seems to be negative. Nov 9 comment Hölder Condition Implying Uniform Convergence I'm working on the exact same problem (Stein and Shakarchi, Complex Analysis, Ch. 3, Problem 5), and I also don't see why this can be so easily asserted. It's a shame no one answered this yet. I attempted to show as $\epsilon\rightarrow 0^+$ that $g(x\pm i\epsilon)$ is uniformly Cauchy, but this didn't get me anywhere either. May 14 comment What are the minimal conditions for the exactness of a 1-form on an open connected subset? Yes - I mean they still need to be connected, but the paths from $\textbf{a}$ to $\textbf{x}$ are only parallel to the axes in some open set about $\textbf{x}$. If $n$ such paths exist and have the same integrals, one for each dimension, then do we have exactness? May 14 comment Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix? @Spacey no - nothing widely used. May 14 comment What are the minimal conditions for the exactness of a 1-form on an open connected subset? I see. Could you please clarify what the minimal condition for exactness is then? It seems that all that's needed is that for every $\textbf{a},\textbf{x}$, that there really only need to be $n$ paths with equivalent integrals, one for each dimension, where the paths don't even need to trace out $\partial\Delta$ fully, but just near $\textbf{x}$. May 14 comment Is there a way in matrix math notation to show the 'flip up-down', and 'flip left-right' of a matrix? For a matrix $M$, will $DM$ and $MD$ suffice for up-down and left-right flips, respectively, where $D$ is the unit anti-diagonal matrix? May 10 comment Why does $\exists x\,\ x = x$? @RickyDemer So, basically, the difference between your formalization and the one in Wikipedia is that there's an additional existential qualifier $\exists\emptyset$ in the Axiom of Infinity? May 7 comment Prove that an open interval and a closed interval are not homeomorphic Right, it should be the induced topology. What I'm saying is that it isn't clear that on the induced topology that the whole unit open interval is not a compact set. On the induced topology it is closed after all. May 7 comment Prove that an open interval and a closed interval are not homeomorphic Why is $(0,1)$ not compact? Its compactness needs to be evaluated relative to its induced topology, not some unrelated superset like $\mathbb{R}$. May 6 comment Prove that an open interval and a closed interval are not homeomorphic Didn't you make assumptions about how the image looks like? May 6 comment How come Stone-Weierstrass theorem does not imply that in a given interval every continuous function has a power series expansion? I'd also note that Stone's extension to the Weierstrass Polynomial Approximation theorem generalizes its conclusions beyond the scope of polynomials anyway. Any point-separating real continuous function algebra on any compact $T_2$ space suffices for the approximation (complex algebras must also be self-adjoint). May 3 comment Show that the functions $m(x) = \inf_{a\leq \xi \leq x}{f(a)}$ and $M(x) = \sup_{a\leq \eta \leq x} {f(\eta)}$ are both continuous from left. The continuity of the function $f$ is a big assumption which was not stated in the question. May 3 comment Show that the functions $m(x) = \inf_{a\leq \xi \leq x}{f(a)}$ and $M(x) = \sup_{a\leq \eta \leq x} {f(\eta)}$ are both continuous from left. If $M(x)$ is not continuous from the left, then $\exists \epsilon >0\forall\delta...$, which is different from what you said. Further, $M$ is monotone increasing, not decreasing. May 3 comment Show that the functions $m(x) = \inf_{a\leq \xi \leq x}{f(a)}$ and $M(x) = \sup_{a\leq \eta \leq x} {f(\eta)}$ are both continuous from left. How would you show that the Heaviside theta is continuous from the left at 1? It is not. May 3 comment Show that the functions $m(x) = \inf_{a\leq \xi \leq x}{f(a)}$ and $M(x) = \sup_{a\leq \eta \leq x} {f(\eta)}$ are both continuous from left. But do you agree that by your definitions for the Heaviside function $M(x)=0$ whenever $x<0$ and $M(1)=1$, so we do not have left-continuity, so the question is ill-formed? May 3 comment Show that the functions $m(x) = \inf_{a\leq \xi \leq x}{f(a)}$ and $M(x) = \sup_{a\leq \eta \leq x} {f(\eta)}$ are both continuous from left. Take $f$ to be the Heaviside Theta on $[-1,1]$, which is 0 for $x <0$ but $1$ for $x\ge 0$, both defined and bounded. It seems that $M(x)=f$, but is clearly not continuous from the left at 1. Perhaps you have flipped the directions? Apr 30 comment A problem on divisibility theorem I do not find this to be a convincing reason why squares and cubes mod 7 all follow this pattern. Jan 29 comment Why does $\exists x\,\ x = x$? @Hurkyl Actually, it seems that this convention may be necessary to the avoid the possibility of an empty model. According to this formulation of the Axiom of Infinity, we must presuppose the existence of $\emptyset$ in order for it to be an element of $\mathbb{N}$. Jan 28 comment Why does $\exists x\,\ x = x$? OK - that answers everything, thanks!