We have $\space \!C = \{f:\mathbb{R} \rightarrow \mathbb{R}\mid f \text{ continuous}\}$. How do I go about proving that it is equinumerous with $\mathbb{R}$?
I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$, the . However, I read in the book &quot;Metric spaces&quot; by Ó Searcóid that set of all $[0, 1]\to\mathbb{R}$ continuous functions is greater than $\mathfrak c$:  &quot;It is demonstrated in many textbooks that $\mathbb{Q}$ is countable, that $\mathbb{R}$ is uncountable, that every non-degenerate interval is uncountable, that the collection of continuous functions deﬁned on  $[0,1]$ is of a greater cardinality than $\mathbb{R}$, and that there are sets of greater and greater cardinality.&quot;  I understand that (via composition with the continuous function $\tan$ or $\arctan$) these sets of continuous functions have the same cardinality. Therefore, which claim is correct, and how do I prove this?
Math people:  The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof.  It is not intuitively clear to me that uncountable ordinals or cardinals should exist.  What axioms of set theory are needed to accept their existence?  I think I have read here that the Axiom of Choice is not necessary.  I am reluctant to accept as "true" any theorems that rely on the Axiom of Choice.  EDIT: This question may be a duplicate: I just found  .  I apologize if this is so. The new profile shouldn't mistakenly imply that you can earn the Informed badge on a meta site I just joined Sports, and I decided to take a look on Sports Meta. I clicked on my profile there. This is part of what I saw:     Yet the profile shouldn't indicate that it is, in theory. . .  This seems to be a bug. How do you improve Drupal performance? Drupal is fast at develop, but the performance is so poor.  It is hard to reach 50 request per second.  And there are so many SQL queries in a simple web page.  If you test the request time of a form submit in Drupal.org it will always take seconds to finish.  How do you improve the speed of your web site? Get the last item in an array Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ See permissions for new files on a given directory I don't understand how my Linux machine is operating on new files.   I have an Amazon Linux AMI (RHEL based distro) and when I execute umask I get 0002, so I get whenever I create new stuff other users won't get write access.  But then I go to my home directory and I type:  $ mkdir myDir $ touch myDir/myFile $ ls -l | grep myDir   and I get  drwxrwxr-x 2 myself myself 4096 May 11 22:37 myDir   and for the folder:  $ ls -l myDir -rw-rw-r-- 1 myself myself 0 May 11 22:37 myFile   So apparently there's more going on there then my umask, since myFile permissions are more restrictive than just write protection.  Digging deeper, if I try:  $ sudo touch /var/run/myPidFile.pid $ ls -l /var/run/ | grep myPidFile.pid -rw-r--r-- 1 root    root       0 May 11 22:42 myPidFile.pid   So myPidFile.pid gets a much more restrictive default permission, under /var/run then myFile gets under my home folder.  We could blame the root umask but if I run umask under root I get 0022 which is indeed more restrictive then my user's 0002 umask but still doesn't explain how the execution bit permission isn't set.  So how can I understand a folder's default permission on Linux? How to rig a shirt with buttons so that the buttons follow the mesh of the shirt I didn't address this in a while because I was busy with something else, but now it has cycled back into my schedule.  Right now I have a shirt modeled and bones set in place. It's already technically rigged, but the only issue is the buttons.   I'm not entirely sure how to go about connecting them to the shirt's deformations. I tried to combine them and the shirt into one object, but because of the shirt's modifiers, (Solidify and subsurf, which I don't want to apply yet), the buttons become extremely deformed. I've tried to match the weight painting intensity where a button is at in correspondence with the shirt, but that doesn't seem to work, maybe it has to do with the specifics of where the button is?   Is there a way to make the buttons follow the shirt's deformations without deforming the mesh vertices of the button? I know that if I probably apply the modifiers on the shirt and combine it with the buttons, it would be okay, but I'm not okay with doing that just yet with the shirt.    I rigged the shirt by deleting all vertex groups, assigning empty groups to the mesh from the armature, and then assigned automatic groups in the weight paint menu.  Is there a way to do this? MATLAB superclass can not be found on the MATLAB's search path Is there a way to kill a process tree using the terminal?  By killing a process tree, I mean killing all descendants of a process along with the process it self.  I tagged this question htop because I am using that utility a lot these days and it would be cool if there was an easy way to kill a process tree inside htop (something better than just selecting all the processes in the subtree). I created some texture maps in MindTex to use in Blender, but I can't figure out how to properly hook them up. The maps I have:  - Diffuse  - Normal  - Height   - Specular   - Gloss   - Self Illumination  - Reflection  - Ambient Occlusion   MindTex (left) vs. Blender (right)  As you can clearly see it's a pretty big difference.  is my material node setup in Blender. In the examples I don't actually use self illumination or reflection images, but including this in the answer would be helpful. How do I use 'chmod' on an NTFS (or FAT32) partition? Spin or rotate an image on hover Is there an invalid pthread_t id? LWC Get Pick-list Field Values I have seen in American Sniper, Dead-7 and Carrie that when someone is killed, they are laying on the ground with blood leaking out of their back or somewhere else on the body.  The puddle continuously grows with few signs of slowing down. This is obviously not CGI because it is a real liquid (may not be real blood as that would spark an emergency in real life).   I have seen in American Sniper that when the protagonist kills yet another enemy, they are filmed hitting the ground and blood starts pooling around their body.  In Carrie: In the dance room where Carrie sets her powers loose and kills a lot of teenagers in the process, there are a few that hit the ground with blood pooling under their bodies. Sometimes they would be on their side with blood coming out.  In Dead 7: Kimono and the other six killed lots of zombies. Some of the six were killed in the process as well before they came to Apocalypta to kill her. They were filmed hitting the ground dead with blood pooling around them.  I am not sure how they film this as they are wearing thin shirts for these scenes, and not only that, but the characters also have blood (obviously fake) only just dripping out of the side of their mouths (an indicator they have a ruptured and wounded stomach or lung (in this case the blood would be pink and frothy).  What is the film technique that allows this to happen? Receiving Push Notifications while in background Why would Mystique's DNA allow for Sentinels that can duplicate mutant powers? Solving an Op-amp circuit The image of the circuit is shown below and it required to find \$V_0\$,  My first attempt at solving this problem is by changing the current source into a voltage source with 1-V and 2k\$\Omega\$ resistance. The fact that the inverting and non-inverting terminals aren't grounded make this problem look difficult.To the point which, I don't know how to proceed with this question or where to start. I would appreciate any help. Following some thought and another schematic from a hint suggested by Alfred, I produced a schematic representing our work.  And  my solution for the problem is below, Using node equation at nodes A and B we have, $$\frac{V_A-V_B}{1k}=-0.5 \text{mA}$$ $$\frac{V_B-(2+V_A)}{1k}=-0.5-x$$ where x is the current that is sent in the output of the op-amp. Using KCL, at the bottom node near the current source we see that the same current that goes through the op-amp also goes through the \$2\text{k}\Omega\$ resistor. Hence, we have, $$\frac{-V_A}{2k}=x$$ Replacing this in the second equation, $$\frac{V_B-(2+V_A)}{1k}=-0.5+\frac{V_A}{2k}$$ And and solving the equations yields \$V_A=-2 \text{-V}\$ and \$V_B=-1.5 \text{-V}\$ "Yesterday" time specification shown when it was actually the day before yesterday Looking at , I see "asked yesterday" (which I really hate; I'd like to see an exact time specification rather than this vague one, especially in the list of questions). But it was not asked yesterday! When I hover over it to show the exact time, it shows it was actually the day before yesterday!  This snapshot was taken at 2012-03-31 15:00 UTC (plus or minus a few minutes):    Whatever different opinions you may have on the implementation, yesterday is a word with a certain meaning. Redefining it is not a good idea. At least rephrase this as "1 day ago" or something. How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?  Edit:  This answer came to mind.  Is it correct?  This answer just came to mind.  By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$.  Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$.  Then $f \notin\{f_n\}_{n \in \mathbb{N}}$.