Let $f:[0,2] \rightarrow \mathbb R$ be a continuous function such that, $f$ is differentiable everywhere on $[0,2]$ except possibly at $x=1.$ If $\lim_{x \rightarrow 1} f~'(x)=2015$. Prove that $f$ is differentiable at $x=1$.  Attempt:  $f:[0,2] \rightarrow \mathbb R$ be a continuous function $\implies ~\exists~m,M \in \mathbb R$ such that $m \le f(x) \le M~\forall x\in [0,2] $  It is given that $\lim_{x\rightarrow 1} f~'(x) = 2015$. We need to prove that $f~'(1) $ exists or that $$ \lim_{h^+ \rightarrow 0 } \dfrac {f(1-h)-f(1)}{h} =  \lim_{h^+ \rightarrow 0 } \dfrac {f(1+h)-f(1)}{h}$$ Could someone please give me a hint on how to move ahead with this problem?  Thank you for your help in this regard.
I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.  I tried applying the Mean Value theorem like this: There exists a $c \in (a, x)$ st $f(x)-f(a)=f'(c)(x-a) \implies \frac{f(x)-f(a)}{x-a}=f'(c) \implies f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{c\to a}f'(c)=A$  However, I think this is wrong. I am pretty sure the Mean Value theorem is needed, but my application of it seems to be incorrect. Would somebody be able to point me in the right direction?
Embedding ordinals in $\mathbb{Q}$ All countable ordinals are embeddable in $\mathbb{Q}$.  For "small" countable ordinals, it is simple to do this explicitly.  $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1\}$.   $\omega*2$ can be done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1+\frac{n}{n+1}:n\in \mathbb{N}\}$, and $\omega*n$ as $\bigcup_{i\le n} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  That immediately also gives $\omega^2$ as  $\bigcup_{i \in \mathbb{N}} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  And going further is still relatively easy: We can biject the above embedding of $\omega^2$ onto a single interval $(0,1)$, e.g. through $f(x)=1-\frac{1}{x+1}$ since this is an order preserving bijection from $\mathbb{Q}^+$ to $(0,1)$, allowing us to get $\omega^2+n$, $\omega^2*n$ etc. And by iterating that process, we can get any ordinal below $\omega^\omega$.   But this sort of embedding fails at $\omega^\omega$ as the iteration of $f$ doesn't seem compatible with taking the infinite union - figuratively speaking, we'd squish things to $0$. So what would an explicit embedding of $\omega^\omega$ look like? And the question, then, is if it is possible to give an explicit embedding of $\epsilon_0$? Is the fact that Peano arithmetic cannot prove well-foundedness of $\epsilon_0$ an indication that it cannot be embedded by such elementary functions and their iterations?  And what about even bigger countable ordinals such as the Veblen ordinals, Feferman-Schütte, Bachmann-Howard? From its definition, even if the above all are possible, I assume that the definitive bound of where one can define an effective procedure for the embedding must be Church–Kleene - is that a correct conclusion?  Lastly, does the situation change when we use $\mathbb{R}$ instead of $\mathbb{Q}$, which would allow us to use, perhaps, more easily understood yet inherently complex functions?  Edited because $\omega^2 \neq \omega^\omega$ Harry Potter: Why 7? While rereading the entire Harry Potter series I noticed that the number 7 occurs quite often, for instance:    Seven books Seven Horcruxes planned by Tom Riddle Seven Weasley kids Seven years at Hogwarts Seven Harry Potters Seven people in a Quidditch team   Did J.K. Rowling ever mention why 7 was so important in Harry Potter? $P(X<Y)$ where X and Y are exponential with means $2$ and $1$ Understanding Group Permissions I'd appreciate help with a couple questions regarding standard file permissions. Assume that I'm not using access control lists.     If a file is associated with multiple groups, must all groups have the same permissions?   If the group permission includes write access, can any member of any group the file is associated with change the file permissions? Can such a person change the file owner?   Say now the file permission is rwxr----- (owner can do everything, but group members can only read). I am in the group associated with the file. I am also the file owner. Which permission am I granted? Read-only, or rwx? I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's equations. These equations are invariant under many transformations (Lorentz transformation, rotations, spatial and temporal translations, etc. By the way is there a way, maybe from group theory, to find all the possible transformations that leaves the equation(s) invariant?) which imply as many conserved quantities thanks to Noether's theorem. 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As far as I know, statements like "IN" or "NOT IN" are even faster with a large amount of values if a subquery on an indexed field without conditions is used.  SELECT * FROM table1 WHERE field1 NOT IN (SELECT index_field FROM table2)   When it comes to use conditions at table2, the query becomes realy slow on a large amount of data.  SELECT * FROM table1 WHERE     field1 NOT IN (         SELECT             index_field         FROM table2         WHERE             user_id = '2'     )   I was thinking about using LEFT Join but when I need to to the filtering for the user_id, it also becomes slow.  How may I solve this problem? Currently, I've no clue about that.  The original query looked sth. like this  SELECT     i.* FROM stream_item si LEFT JOIN user__publisher_item ui     ON ui.user_id = 2     &amp;&amp; ui.item_id = si.item_id INNER JOIN item i     ON i.id = si.item_id WHERE      si.stream_id IN (5,7,8,9,19,24,29,42,43,44)     &amp;&amp; ui.id IS NULL || (         ui.`read` = 0 || ui.saved = 1     ) GROUP BY     si.item_id ORDER BY     si.`found` DESC LIMIT 0,10     id      select_type     table   type        possible_keys                                                                           key                 key_len     ref                 rows        Extra 1       SIMPLE          si      index       PRIMARY,stream_2_item,stream_id_found                                                   stream_2_item       4           \N                  663236      Using temporary; Using filesort 1       SIMPLE          ui      eq_ref      user_id_item_id,user_2_item,user_id_read_saved_hidden,user_id_saved,user_id_hidden      user_id_item_id     8           const,si.item_id    1           Using where 1       SIMPLE          i       eq_ref      PRIMARY                                                                                 PRIMARY             4           si.item_id          1      I have a table which represents the n:m relationship between stream and items. 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