prove that $f$ is continuous over $[0,1]\cap\mathbf{Q}^{c}$
Proof of continuity of Thomae Function at irrationals.
How to split a very long equation (using AMSMATH package)? Word with the same meaning as "good moments" or "pleasant memories"? Is there a single word with the same meaning as "good moments" or "pleasant memories"? I'm looking for a word which describes good thing(s) that happened to someone. The only word I can think of is "joy".  Some examples might be:   Wayne has many pleasant memories of the days spent with the family. She could remember many good moments from the last weekend. Can you take the same feat multiple times? Is there a way to override class variables in Java? Normality of dependent variable = normality of residuals? This issue seems to rear its ugly head all the time, and I'm trying to decapitate it for my own understanding of statistics (and sanity!).   The assumptions of general linear models (t-test, ANOVA, regression etc.) include the "assumption of normality", but I have found this is rarely described clearly.   I often come across statistics textbooks / manuals / etc. simply stating that the "assumption of normality" applies to each group (i.e., categorical X variables), and we should we examining departures from normality for each group.   Questions:     does the assumption refer to the values of Y or the residuals of Y?  for a particular group, is it possible to have a strongly non-normal distribution of Y values (e.g., skewed) BUT an approximately (or at least more normal) distribution of residuals of Y?   Other sources describe that the assumption pertains to the residuals of the model (in cases where there are groups, e.g. t-tests / ANOVA), and we should be examining departures of normality of these residuals (i.e., only one Q-Q plot/test to run).  does normality of residuals for the model imply normality of residuals for the groups? In other words, should we just examine the model residuals (contrary to instructions in many texts)?  To put this in a context, consider this hypothetical example:   I want to compare tree height (Y) between two populations (X).   In one population the distribution of Y is strongly right-skewed (i.e., most trees short, very few tall), while the other is virtually normal  Height is higher overall in the normally distributed population (suggesting there may be a 'real' difference). Transformation of the data does not substantially improve the distribution of the first    population.  Firstly, is it valid to compare the groups given the radically different height distributions? How do I approach the "assumption of normality" here? Recall height in one population is not normally distributed. Do I examine residuals for both populations separately OR residuals for the model (t-test)?     Please refer to questions by number in replies, experience has shown me people get lost or sidetracked easily (especially me!). Keep in mind I am not a statistician; though I have a reasonably conceptual (i.e., not technical!) understanding of statistics.  P.S., I have searched the archives and read the following threads which have not cemented my understanding: Simple approximation of Inverse Incomplete gamma function Pointwise convergence of $h_n(x) = x^{1+\frac{1}{2n-1}}$ Okay, maybe I'm missing something obvious but consider $h_n(x) = x^{ 1+ \frac{1}{2n-1}}$ on the set $[-1,1]$. Now consider a fixed $x \in [-1,1]$, we have:  $$\lim_{n\rightarrow \infty} h_n(x) = x \lim_{n \rightarrow \infty} x^\frac{1}{2n-1} = x$$  But my book says it's $|x|$? Why? Irreducible polynomials and affine variety Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite. $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki How can I do a line break (line continuation)? Why is the Principle of Superposition true in EM? Does it hold more generally? In the theory of electromagnetism (EM), why is the principle of superposition true? Can we read it off from Maxwell's equations directly?  Does it have any limit of applicability or is it a fundamental property of nature? Group member cannot write even with write permission Attempted multiple solutions for malwares like androidSystem, timeService and monkeyTest, but to no avail Proof on Catalan numbers What are the advantages of a custom ROM over rooted stock? How do I end a Tkinter program? Let's say I have this code:  from Tkinter import *  def quit():     # code to exit  root = Tk() Button(root, text="Quit", command=quit).pack() root.mainloop()   How should I define the quit function to exit my application? How to efficiently find positions of duplicates? Strange arrow mark with TikZ edge and anchors. When I draw an edge between two nodes with anchors and globally set option [-&gt;], I get an additional arrow tip symbol at the beginning of the last edge in the path. For example in  \documentclass{article} \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \node (A) at (0,0) {$A$};     \node (B) at (1,0) {$B$};      \draw[-&gt;] (A.north east) edge (B.north west);      \draw (A.south east) edge[-&gt;] (B.south west); \end{tikzpicture} \end{document}   the first \draw command produces incorrect output, while the second produces correct output:    Am I doing something wrong, or is this a bug in TikZ? I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance. LaTeX doesn't recognize "et al." in the Bibliography I'm using Jabref to manage my Bibliography.  Sometimes, I put directly:     First Author et al.   when adding the article author's name in Jabref.  When compiling my .tex file, LaTeX doesn't recognize the et. al as more names. It takes it as a name/surname.  This is an example of a Bibliography entry in Jabref.  @ARTICLE{Wolf2003,     author = {Wolf, K.et al.},     title = {Compensation mechanism in tumor cell migration: mesenchymal–amoeboid     transition after blocking of pericellular proteolysis},     journal = {J. Cell Biol.},     year = {2003},     volume = {160},     pages = {267–277},    }   This is a .texMWE with the bib entry mentioned:  \documentclass[11pt]{book}   \usepackage{cite}   \bibliographystyle{unsrt} \begin{document}   Here is the bib entry mentioned\cite{Wolf2003}    \bibliography{bibfile}  \end{document}   Here's the output:   How can I solve this problem?