Show that if $\int_{a}^{b}g(t)dt=0$ then $g(t)=0$ for all $t\in[a,b]$
If integral is zero and function is continuous and non negative, then what about the function?
Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context. Is there a Stack Exchange site for both technical and behavioural interview questions? Where I could ask questions about, for example:   Sample questions and solutions, experiences, or seeking advice for interviews A place where I could ask interview questions and what the best way to answer it (not just a correct way or the most efficient).   What to do in a certain situation such as negotiation, unfair interviews, etc.   Experiences interviewing at so-and-so company. "With who" vs. "with whom" Is this correct?     The person with whom I'm doing the project should be here soon.   If it is, is with always a dative preposition (like mit in German)? If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Why do we call a 1/2 spin particle satisfying the Dirac equation a , and not a  or a ? Why is method overloading and overriding needed in java? Recursively copy files from one directory to another I have all my music in a folder /media/kalenpw/MyBook/Music/ABunchOfOtherFoldersWithFilesInside. I want to copy all of the mp3s to /media/kalenpw/HDD/Music so I used:  cp -R /media/kalenpw/MyBook/Music/*.mp3 /media/kalenpw/HDD/Music   however this only copied the mp3s in the root music folder and did not open any of the artist subdirectories and copy those files.  I was under the impression -R would recursively copy all the files. How can I achieve said goal? Why does an environment's label have to appear after the caption? How to show that a Banach space $X$ is reflexive if and only if its dual $X'$ is reflexive? Proof: Every convergent sequence of real numbers is bounded Selectively coloring text in RichTextBox How can I paint in red every time I meet the letter "A" in RichTextBox? The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon? I would like to know if there is any way to add custom behaviour to the auto property get/set methods.  An obvious case I can think of is wanting every set property method to call on any PropertyChanged event handlers as part of a System.ComponentModel.INotifyPropertyChanged implementation.  This would allow a class to have numerous properties that can be observed, where each property is defined using auto property syntax.  Basically I'm wondering if there is anything similar to either a get/set template or post get/set hook with class scope.  (I know the same end functionality can easily be achieved in slightly more verbose ways - I just hate duplication of a pattern) Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $ Let $f:\Bbb R\to (0,\infty)$ be a differentiable function such that for some constant $k$ we have, $$|f'(x)-f'(y)|\le k|x-y|$$   for all $x,y \in\Bbb R.$ Then prove that, $$(f'(x))^2&lt;2 kf(x).$$   Due to lack of arguments I could not proof this inequality. But rather I proved that is true for the particular function,   $$f(x) =\cos^2(x)+1\implies f'(x) = -\sin(2x).$$ And we readily have,$$|\sin(2x)-\sin(2y)| = \left|\int_{2x}^{2y}\cos t dt\right|\le 2|x-y|$$ as well   $$(f'(x))^2 = 4 \sin^2 x\cos^2x &lt; 4(\cos^2x +1) = 4f(x).$$  It is also true for the function $x\mapsto \sin^2x +1.$   From these example I don't see how to prove the general case. Any hint will be welcome. How to find regression coefficients $\beta$ in ridge regression? In ridge regression, the objective function to be minimized is: $$\text{RSS}+\lambda \sum\beta_j^2.$$   Can this be optimized using the Lagrange multiplier method? Or is it straight differentiation? How can I use a variable in the replacement side of the Perl substitution operator? Why does time change in Ubuntu after installing Windows Today I was installing Windows 7 on the same computer I have Ubuntu. For some reason, after installing Windows 7 and going back to Ubuntu the time went from the correct hour at that moment, 12:30PM to 7:30AM. This happened after installing Windows 7. For what I understand the BIOS has a time set, Ubuntu has the same time that is shown in the BIOS when installing it. At least up to this point is what I have seen. Now after installing Windows 7 (And believe me this is only for testing the performance of Windows 7 against Ubuntu in which case Windows 7 is falling behind a lot on the same hardware) the time changes in Ubuntu to the wrong one that is set in Windows 7.  I actually do not change the time in Windows 7 since this is one of the checks I am doing to see if the operating system can correctly grab the time. Windows 7 fails. Ubuntu does not BUT Windows 7 actually changes the time for Ubuntu if I boot with it afterwards.  So my question is, why/how does time change in Ubuntu after installing Windows.  NOTE - I have set Ubuntu time to "manual" instead of grabbing it from the Internet just to test out this. How do I know if food left at room temperature is still safe to eat? If I left food out of the refrigerator for some period of time, is it still safe? If I left it out too long, can I salvage it by cooking it more? What does ':' (colon) do in JavaScript? New review navigation does not tell review-banned users what happened or why When I am review-banned, the direct effect is that when I click the review icon in the top-bar, I get this:        This doesn’t tell me anything about what's going on, just that all review queues have miraculously become unavailable to me. It doesn't show that it's because I've been banned, or why I've been banned, whether it's for failing audits or whether there is a custom message supplied by a moderator. Only when I new-tab-click that button, it takes me to /review, or when I manually navigate there, I get :        This is confusing for review-banned users who use the dropdown in their standard workflow (). I only performed the above test to reproduce the experience of a confused review-banned user on a site I moderate.  Please replace the dropdown message with something more informative in case of a review ban.