Not really sure how to proceed with this one. Can anyone help?  The question is:     Show that any continuous map $f: \mathbb{R}\to\mathbb{Z}$ is constant.
I'm stuck on a question which asks to prove that any continuous integer-valued function of a real variable is constant. In my lecture notes we are told that a map is continuous if the preimage of any open set is open.  How do I use this to answer the question? A full answer would be greatly appreciated as I can't seem to figure out how to solve it in my  notes, in analysis books or anywhere online.
Which site deals with career development questions for programmers? Is there any site in the network where career development questions about the programming profession can be asked ? I thought Programmer's was the site for it, but that doesn't appear to be the case. How do I calculate n modulo 3 in LaTeX? I have an app installed on my phone, which I've installed from the Google Play Store.  If I update the app and don't like the upgraded version, is there a way to undo the update and downgrade back to the version of the app that I had previously installed?  I tried searching a bunch on this site, and I found lots of questions about downgrading the OS/ROM/firmware, but nothing about downgrading app versions or reverting an app update. Deploy a project using Git push How to move apps from internal to external storage? Rendering of Favorite/Ignored Tags in Firefox I have spotted a bug in rendering of Favorite Tags and Ignored Tags section in Firefox. This rendering issue is present on both stackoverflow.com and meta.stackoverflow.com sites.  I'm using Firefox 17.0.1 on Windows 7 x64. In Google Chrome page is rendered correctly.  Screenshots:  Firefox (please note cropped right part of both Add buttons)    Chrome (buttons are rendered correctly) Javascript to sort contents of select element is there a quick way to sort the items of a select element? Or I have to resort to writing javascript?  Please any ideas.  &lt;select size="4" name="lstALL" multiple="multiple" id="lstALL" tabindex="12" style="font-size:XX-Small;height:95%;width:100%;"&gt; &lt;option value="0"&gt; XXX&lt;/option&gt; &lt;option value="1203"&gt;ABC&lt;/option&gt; &lt;option value="1013"&gt;MMM&lt;/option&gt; &lt;/select&gt; Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$ I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$  I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements from $n$ and then $i$ elements from those $k$, so $\displaystyle \sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i}$ is counting the number of ways to do that $\forall k$.  I know that $\displaystyle \binom{n}{i} = \binom{n}{n-i}$ and that $2^{n-i}$ is the number of elements in the power set for $[n]\setminus[i]$.  How do I show that both sides are equivalent? I'm stumped on this one. Could anyone give me a pointer in the right direction? I built a desktop computer a couple years back with the following specs:  CPU: Intel  Yorkfield 2.5GHz 6 MB L2 Cache LGA 775 95W Quad-Core Processor BX80580Q9300 Motherboard:  LGA 775 NVIDIA nForce 680i SLI ATX Intel Motherboard Video Card: Two  GeForce 8600GT SCC 256 MB 128-bit GDDR3 PCI Express x16 SLI Supported Video Card PSU: SeaSonic S12 Energy Plus  ATX12V V2.3 / EPS12V V2.91 SLI Certified CrossFire Ready 80 PLUS Certified Active PFC Power Supply Memory: Two G.SKILL 4 GB (2 x 2 GB) 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400) Dual Channel Kit Desktop Memory   Since its inception, the machine has periodically locked up, the regularity having varied over the years from once a day to once a month. Typically, lockups happen once every few days. By &quot;lockup&quot; I mean my computer just freezes. The screen locks up, I can't move the mouse. Hitting keys on my keyboard that normally turn LEDs on or off on the keyboard (such as Caps Lock) no longer turn the LEDs on or off. If there was music playing at the time of the lockup, noise keeps coming out of the speakers, but it's just the current frequency/note that plays indefinitely. There is no BSOD. When such a lockup occurs I have to do a hard reboot by either turning off the computer or hitting the reset button. I have the most recent version of the NVIDIA hardware drivers, and update them semi-regularly, but that hasn't seemed to help. I am currently using Windows 7 x64, but was previously using Windows Server 2003 x64 and having the same lockup issues. My guess is that it's somehow video driver or motherboard related, but I don't know how to go about diagnosing this problem to narrow down which of the two is the culprit.  Additional information re: cooling Regarding cooling... I've not installed any after-market cooling systems aside from two regular fans I scavenged from an older computer. The fan atop the CPU is the one that shipped with it. One of the two scavenged fans I added it located at the bottom tower of the corner, in an attempt to create some airflow from front to back. The second fan is pointed directly at the two video cards.  SpeedFan installation and readings Per studiohack's suggestion, I installed , which provided the following temperature readings:  GPU: 63C GPU: 65C System: 76C CPU: 64C AUX: 36C Core 0: 78C Core 1: 76C Core 2: 79C Core 3: 79C   Update #3: Another Lockup :-( Well, I had another lockup last night. :-( SpeedFan reported the CPU temp at 38 C when it happened, and there was no spike in temperature leading up to the freeze. One thing I notice is that the freeze seems more likely to happen if I am watching a video. In fact, of the last 5 freezes over the past month, 4 of them have been while watching a video on Flickr. Not necessarily the same video, but a video nevertheless. I don't know if this is just coincidence or if it means anything. (As an aside, each night before bedtime my 2 year old daughter sits on my lap and watches some home videos on Flickr and, in the last month, has learned the phrase, &quot;Uh oh, computer broke.&quot;)  Update #4: MemTest86 and 3DMark06 Test Results: Per suggestions in the comments, I ran the MemTest86 overnight and it cycled through the 8 GB of memory 5 times without error. I also ran the 3DMark06 test without a problem (see my scores at ). So... what now? :-) Any further suggestions on what to check? Is there some way to get a stack trace or something when the computer locks like that? Resolution I have never did figure out the particular problems, but based on the suggestions here and elsewhere, I'm presuming it was a motherboard issue. In any event, I recently upgraded my system, buying a new motherbeard, PSU, CPU, and RAM, and that new rig has been working splendidly the past several weeks. I am using the same graphic cards as in the old setup, so I think it's safe to reason that they weren't the cause of the problem. How do I identify SMD components? (or how do I identify any component) How do I identify the markings on an SMT component and match it up with a part number so I can be a good designer and actually look at a datasheet (and read the whole thing)? Or identify a part to replace an unknown part on a PCB? Are proofs by contradiction really logical? Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.   My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?  That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true". Is it possible to scale up an image to increase the the pixel density? In  (2017), Logan seems to be dying from sort sort of ailment, that prevents him to heal properly and seems to be decaying his body in a way that wasn't happening in any of the other films his character appears in.  I don't recall that this is being explained in the film. The Wikipedia page only mentions "adamantium poisoning" and "his healing factor fading", but it is not backed up with citations, so I would love to get an in-depth answer for this question, maybe using excerpts from the comic or original source material.  What is happening to Logan? Wikipedia says that the name of concept comes from physics, but I cannot find any similarity between these two concepts. I've only seen The Force Awakens once, but I'm as far as I can remember, Phasma doesn't really do much, except:      get taken hostage that one time and disable the shields around the Starkiller after being asked twice   Was she originally supposed to play a bigger role?  I was really expecting to see much more of her, or at least see her in combat. Would it help if you jump inside a free falling elevator? Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When? This is an exercise from textbook Analysis I by Amann/Escher.  Let $G$ be a group and $X$ a nonempty set. Then $G$ acts (from the left) on $X$ if there is a function $$G \times X \rightarrow X, \quad(g, x) \mapsto g \cdot x$$ such that: GA1: $e \cdot x=x$ for all $x \in X$. GA2: $g \cdot(h \cdot x)=(g h) \cdot x$ for all $g, h \in G$ and $x \in X$. Let $\mathrm{S}_{m}$ be the set of all permutations of $\{1,2,\cdots,m\}$. Then $(\mathrm{S}_{m},\circ)$ is a group where $\circ$ is function composition operation. Show that $$\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}, \quad(\sigma, \alpha) \mapsto \sigma \cdot \alpha := \left(\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(m)}\right)$$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$.  I am stuck at proving the GA2, i.e. $\pi \cdot (\sigma \cdot \alpha) = (\pi \circ \sigma) \cdot \alpha$.  My attempt: Clearly, $\operatorname{id}$ is the identity element of $\mathrm{S}_{m}$ and $\sigma \cdot \alpha = \alpha \circ \sigma$. Because $\operatorname{id} \cdot \alpha = \left(\alpha_{\operatorname{id}(1)}, \ldots, \alpha_{\operatorname{id}(m)}\right) = \left(\alpha_1,\cdots,\alpha_m\right) = \alpha$, GA1 follows. For $\pi,\sigma \in \mathrm{S}_{m}$ and $\alpha \in \mathbb{N}^{m}$, we have $$\pi \cdot (\sigma \cdot \alpha) = (\alpha \circ \sigma) \circ \pi = \alpha \circ (\sigma \circ \pi) = (\sigma \circ \pi) \cdot \alpha$$  It is clear that $\sigma \circ \pi = \pi \circ \sigma$ if and only if $(\mathrm{S}_{m},\circ)$ is Abelian, which is not true in general. Please help me figure out where I am wrong! Workflow for converting LaTeX into Open Office / MS Word Format I often have to write up reports based on the analysis of some data. I use R to analyse the data and export tables, figures, and text. This is then included into a LaTeX document either using input or Sweave ().  However, when I collaborate with others, I sometimes need to provide a document in Open Office / MS Word format.  Question:  Thus, assume the simplest scenario   I have a LaTeX document with text, tables, and figures I need to export this reliably into Open Office or MS Word format: this includes mathematical formulas, table formatting, and quality figures) I don't need to go back from MS Word to LaTeX   What is a reliable, efficient, and preferably free process?  Initial Thoughts:   Export to HTML,  LaTeX - LyX - ODF Use odfWeave  that deals with some specific issues associated with LaTeX to Open Office conversion    I was hoping that there is an expert out there who has worked out a good system already. AES and DES - reusing the same round keys How to kill all process with given name?