Hagen von Eitzen
Reputation
168,527
99/100 score
 3h comment How to prove that $\sin \left(\frac 1 {x-1}\right)$ is discontinuous at $1$? You mean $x=1$, not $x=0$. The OP function is deliberately translated by $1$ for some reason 3h comment How to prove that $\sin \left(\frac 1 {x-1}\right)$ is discontinuous at $1$? @DanielHolmes I would reread the definition of "function is continuous at" if I were you and check wether it requires "is defined at" as prerequisite. -- That being said, could it be that you simply ask whether $\lim_{x\to 1}\sin\frac1{x-1}$ exists? 3h comment Integral of product of $h_{x_{i}}(x)$ and $h_{x_{j}}(x)$, where $h_{c}(x) = \min(c,x)'$ Can you express $h_c(x)$ differently? What is $h_c(x)$ for $xc$? 5h comment Find the limit $\lim_\limits{x\to 0^+}{\left( e^{\frac{1}{\sin x}}-e^{\frac{1}{x}}\right)}$ Can you use the Mean Value Theorem? 5h comment Set characterization @user138364 My equality does not contradict $\frac13\in A$, does it? 1d comment A set contains $0$, $1$, and all averages of any subset of its element, prove it contains all rational number between 0 and 1 @AitorOrmazabal Power set, presumably 1d comment Proof-verification: $\forall a \space \exists! r\in${0,…,n-1} with $a \equiv r\space (mod \space n)$ For one, you only have $s\in \Bbb Z$. Unless you assume (wlog.) that $r'\ge r$. - But most importantly you only show that there is at most one such $r$ when the task is to show that there is exactly one such $r$. 1d comment Why not define the Conway base-5 function, instead of base-13? @MarcelT. With sufficiently sophisticated encoding one might even come up with base 3 encoding, e.g. if $x$ in base $3$ after removal of the period matches the pattern $[012]^*2\underbrace{[01]}_a\underbrace{[01]^*}_b2\underbrace{[01]^*}_c$ (with the last Kleene star actually meaning infinitely many digits), we might map this number to $(-1)^a\cdot2^y\cdot z$ where $y$ is the natural number represented in binary by $b$ and $z$ is the real number represented in binary by $0.c$; you could surely even formulate a base 2 only method, but whatfor? The base-13 explanation is just less convoluted 1d comment How to prove one angle is not 60 degree However, the error in $65\ne 64$ is quite small. This tells us that $\frac 74$ is a quite good approximation of $\sqrt 3$. 2d comment If $x^m + y^m = z^m$ has no solutions and $m \mid n$, then $x^n + y^n = z^n$ has no solutions How is it obvious? Would it also be obvious without $m\mid n$? IIf no, you should be able to formulate why the former is obvious, arriving at a proof. 2d comment open cover definition of compactness - technicality I don't understand Provided that $A$ is compact, the answer is yes. That is exactly what the definition says. Or more precisely, the definition requires only $\subseteq$ and guarantees only $\subseteq$, but as you in fact found an open cover with $=$ and the finite subcover cannot be larger, you also get $=$ 2d comment Sorting almost sorted array in $O(n)$ time You really mean final position, i.e., $2,3,4,5,6,7,8,9,1$ is not almost sorted? Nov 23 comment How do I develop numerical routines for the evaluation of my own special functions? Well-behaved functions should allow nice approximations by polynomials (piecewise). However for the desired precision this may be challenging. On the other hand, the surrounding algorithm might allow you to work with less precision most of the time, e.g., to solve $f(x)=b$ by binary search you will not need much precision as long as you are not yet close. Nov 23 comment Example of Category, covariant functors Why does this ask for well-defined (as opposed to defined or just "is a category")? The term "well-defined" is usually only used for cases where a definition a priori depends on some choice made but turns out to actually not depend on that choice. And I don't see any choices made here ... Nov 23 comment Does there exist $n$ such that all numbers $n,2n,\dots,2000n$ have the same digits? @mvw Darn, why would you even check such a thing? :) Nov 23 comment Does there exist $n$ such that all numbers $n,2n,\dots,2000n$ have the same digits? @fleablood The problem with $2000n$ is not harder than the problem with $2n$ because of $2000n$. Rather, it is that $1999n$ must also have the right digits ... Nov 23 comment Proof of Riemann Hypothesis Already the very first sentence of the introduction has two mistakes in "every nontirivial zeros" Nov 23 comment Prove limit superior of a sequence is greater than or equal to limit superior of the Cesàro mean of that sequence It is not even clear to me that $\lim\sum_{i=0}^n\frac{x_n}n$ exists. Do you mean $\limsup\sum_{i=0}^n\frac{x_n}n$? Nov 22 comment Suppose $f : [a, b] \to \mathbb{R}$ is bounded and $f \in \mathcal{R}[c, b]$ for all $a < c < b$. Show $f \in \mathcal{R}[a, b]$. What does the notation R[c,b] stand for? Space of functions that are Riemann integrable on $[c,b]$? Nov 22 comment Proving 3 functions combined are injective Do you already know that the combination of two injectives is injective?