Let $f:\mathbb{R}\rightarrow\bar{\mathbb{R}}$. Prove for $C(f)=\{x\in\mathbb{R} :f \text{ is continuous in }x\}$ is a $G_\delta$
How to show that the set of points of continuity is a $G_{\delta}$
I was just wondering whether the following statement is correct.  Let R be a ring and M a noetherian R module. Then M is finitely generated. I (as a middle-school student) was wondering how virtual particles even conserve energy of the entire system? I don't mean just the particle's energy, but conservation with respect to the surroundings?  Let me explain more, Einstein proved that $E = mc^2$ and equivalently, we can say $m = \frac{E}{c^2}$ . We however know mass causes curvature in space-time (gravity well). Now that statement being completely valid we can use Quantum Fluctuation, implying energy and time can be related by the relation:  $\Delta{E}\Delta{t}\approx\frac{\hbar}{2}$ and by that simple relation we can make the following points:   As the energy increases, the time allowed to be "borrowed" decreases, before it gives it back to the universe. As time increases, the energy allowed to be "borrowed" decreases.    These are completely satisfactory points I've made, I believe.  Now, I have read that the vacuum is filled with constantly bubbling virtual particles that get created from the energy "borrowed" by the above relation and is destroyed again, which satisfies the energy conservation laws.   This seems to imply that  in the short time the virtual particle exists, they must have a very small mass which is: $m_{1} = \frac{\Delta{E}}{c^2}$ and using basic classical mechanics rather than GR (for sake of simplicity) we calculate that the gravitational force mediated by this short-lived virtual particle;  $F_{g}= G\frac{m_1 * m_x}{r^2}$   where $m_x$ is an particle\mass in question. Nevertheless since we know the force set upon an object moves the object in a direction, we can say it would increase its kinetic energy by this factor: $F_g = \frac{1}{2}mv^2$ which is equal to $G\frac{m_1 * m_x}{r^2} = \frac{1}{2}mv^2$.  Surely, this implies that every time quantum fluctuations happen, energy of real-particles\masses increases and therefore violates energy conservation, but what prevents this violation from taking place?  If virtual particles are not really present and interact with objects like I thought, how can Hawking radiation (currently a large concept in black hole physics) then be accepted in physics community? Having Overfull and underfull box warnings. Please help out as there are many matrices and equations Moving an already-running process to Screen How to compute $\lim_{n\rightarrow\infty}e^{-n}\left(1+n+\frac{n^2}{2!}\cdots+\frac{n^n}{n!}\right)$ There is a probabilistic method to solve it. But I am not familiar with probability. I am trying to compute it by analytic method, such as using L Hospital's rule or Stolz formula, but they are not working. How do I submit a form via AJAX? 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It's caused entirely by the extra variables.  If it helps anything, they're "name", "email" and "message". No human players, just AI and have the ability to scroll around and see what the AIs are doing with the turn management being also automated, so the AIs just end turn when they are done. Oh, and all of this hopefully without any sort of fog of war.  This way, I can see how the AI manages resources and units and maybe learn a thing or two as a civ newb. When to use ' (or quote) in Lisp? Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. Suppose that $f(x)$ is continuous on $(0, +\infty)$ such that for all $x &gt; 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function. My attempt is to show that for any point $a \neq b$ , we have $f(a)=f(b)$. But I have no idea on how to get this. Anyone can help? How much energy does lowering an object into a black hole generate? An object of mass m is slowly lowered into a black hole of mass 1000 m. Is the amount of braking energy larger than $0.6 mc^2$?   Now what if, after lowering the mass close to the event horizon, we wait the black hole to shrink to mass 100 m, by emitting Hawking radiation, and then we lower the mass some more? Is the braking energy of the second lowering process larger than $0.5 mc^2$?   I wonder can we extract more than $E=mc^2$ energy from mass m?  (In this thought experiment we use a wire and a winch, or anything that works) For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$ Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally:     $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$   If this statement is true, what is the proof ? Teams of users rails app breaking of output into new lines True Ubuntu on Chromebook ARM