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 Apr20 comment Problem about $\sin(1/x)$ in topology. (open and closed functions) I also do not understand your purported argument as to why $f$ is continuous. Isn't it much easier to argue that $f$ is the composition of continuous maps? Apr20 comment Problem about $\sin(1/x)$ in topology. (open and closed functions) @AlexR: Exhibiting an open subset $A$ of $(0,\infty)$ such that $f(A) = [-1,1]$ does not show that $f$ isn't an open mapping since $[-1,1]$ is open in the codomain. Apr20 revised Problem about $\sin(1/x)$ in topology. (open and closed functions) less-than-or-equals is not written as "<=" in (La)TeX Apr20 comment Is it true that if $g,h\in G$ have order $p$, where $p$ is prime, the only possible order of $\langle g\rangle \cap \langle h \rangle$ is $p$? $$o(x) \mid \lvert \cdots \rvert$$ may not be as bad as the legendary $$\Xi \over \bar\Xi$$ but I do still think that it deserves a good old "your notation sucks!" May I suggest using $\#$ for "number of elements of" next time? Apr14 revised What's the Lebesgue measure of this set? added 227 characters in body Apr14 answered What's the Lebesgue measure of this set? Apr10 answered Proving $\int_a^b f = - \int_b^a f$ from the reflection property of the integral? Apr7 comment Why are vector valued functions 'well-defined' when multivalued functions aren't? It seems that you missed my point. I was saying that it isn't clear from $G$ whether I was talking about $x \mapsto x^2$ as a function from $\mathbb R$ to $\mathbb R$ or as a function from $\mathbb R$ to $[0,\infty)$. It's clearly only surjective in the latter case. Expanding the codomain of a function doesn't change its value, behaviour or graph (in analysis we most of the time don't really care about specifying the codomain too precisely, even), but it does change it as a function since it may go from being surjective to being... well... not. Apr6 comment Why are vector valued functions 'well-defined' when multivalued functions aren't? Indeed, the Bourbakist definition of (partial) function is as an ordered triple $f = \langle D,G,C\rangle$ "(domain, graph, codomain)" such that $$G \subseteq D \times C \text{ and }(x,y),(x,y') \in G \implies y=y'.$$ A function is then a special case of partial function where $$x \in D \implies (\exists y \in C : (x,y) \in G).$$ Apr6 comment Why are vector valued functions 'well-defined' when multivalued functions aren't? If we're going to be pedantic, then the "a function is its graph" definition is just barely not good enough because it doesn't allow one to talk of surjective functions. For example we could look at $$G = \{(x,y) \in \mathbb R^2 \mid y = x^2\} = \{(x,y) \in \mathbb R \times [0,\infty) \mid y = x^2\}.$$ Is $G$ the graph of $\mathbb R \owns x \mapsto x^2 \in \mathbb R$, or of $\mathbb R \owns x \mapsto x^2 \in [0,\infty)$ or maybe it has some entirely different codomain? Mar30 revised Computing the Laplace transform of $\frac {f(x)}{x}$ added 312 characters in body Mar30 comment Computing the Laplace transform of $\frac {f(x)}{x}$ I'm not sure exactly what you're asking, but you're definitely doing the integral wrong, because you can get negative results from putting non-negative functions into what you've computed. Mar29 answered Computing the Laplace transform of $\frac {f(x)}{x}$ Mar29 revised Computing the Laplace transform of $\frac {f(x)}{x}$ added 11 characters in body; edited title Mar28 comment Dual space of a closed subspace of a Hilbert space It should be noted though, that $\mathcal D'(\Omega)$ is a bit special because one usually gives it a different (finer) topology than the obvious one it has as the dual space of $C_c^\infty(\Omega)$. Mar27 comment Weakly convergence in $W^{1,p}_0$ and strong convergence in $L^p$ For reference: $a: \Omega \times \mathbb R \to \mathbb R$ is a Carathéodory function iff for all $(x,z)$ in its domain $a(x,\cdot)$ is continuous and $a(\cdot,z)$ is measurable. Mar27 revised A proposition of Urysohn's Lemma in real analysis added 31 characters in body Mar27 comment A proposition of Urysohn's Lemma in real analysis TeX tip: The usual way of writing the support of a function $\varphi$ is "$\operatorname{supp} \varphi$" (\operatorname{supp} \varphi). If one is writing an actual TeX document (rather than using MathJax on this or some other site) then one would use \DeclareMathOperator in the preamble to define \supp as a new math operator. Mar27 comment What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$? TeX tips: Do not use \bf Z. Instead use \mathbb Z to get the usual symbol for the integers. I also see that you forgot a \text in the \underbrace construction. Mar27 comment What is the cardinality of the set of all functions from $\mathbb{Z} \to \mathbb{Z}$? $\mathbb Z^{\mathbb Z}$ (your $A$) is not a superset of $2^{\mathbb Z} = \{0,1\}^{\mathbb Z}$ (your $B$) in the usual representation of functions as sets in ZF. Furthermore, why are you writing $\mathcal N$ for the cardinality of $\mathbb R$ instead of the usual $\mathfrak c$, and $\mathcal N_0$ instead of $\aleph_0$?