Because Apostol just defined $\inf f$ and $\sup f$, and is now talking about the maximum and minimum of $f$ over $a,b$ and calling them $M(f)$ and $m(f)$
Let $\Gamma = SL_2(\mathbb{Z})$.
Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$.
Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$.
Let $\alpha = \left( \begin{array}{ccc}
p & q \\
r & s \end{array} \right)$ be an element of $\Gamma$.
We write $f^\alpha(x, y) =...
@JohnSenior "Thank you for the offer Sir. I am sorry to inform you that I have lost the kit, but I have attached the poo to this letter anyway. I hope this is okay too. Sincerely, John"
@JonasTeuwen Take a theorem you like and a continuous function on $[a,b]$. Suppose the theorem is not true on $[a,b]$. Then it must not be true in either $[a,c]$ or $[c,b]$ (or both) where $c$ is the mid point of the interval. Denote this interval where the theorem fails with $[a_1,b_1]$ (if the theorem fails on both, choose the left interval). Repeat the process designing by $[a_{n+1},b_{n+1}]$ the half of $[a_n,b_]$ where the theorem is false.
Now that we're in a $(b-a)/2^n$ scale we can argue by some $\epsilon-\delta$ proof to show the theorem is indeed true, so that BOOM, it is now true over all $[a,b]$
@PeterTamaroff I think I will just by the cheapest kindle 69USD, I have iPad but I want some faster way of getting course books to lessons...instead of waiting them from post
@JonasTeuwen For example, you can use that to prove the Bolzano Weiertrass , uniform continuity, the theorem that every continuous functions achieves a maximum and a minimum
i am asked, which of the following implications is false? "2+2=4 only if -1 is a positive number" or "if 3x3=9 then -1=1" aren't they both false implications?
(first one is p only if q, second is if p then q, both p -> q, both p is true and q is false)
from that site: "Only if the fruit is an apple, will Madison eat it." or "Madison will eat the fruit only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → fruit is an apple")
notice "only if" comes right before the conclusion, not the hypothesis
Later you have quantifiers that builds on top of propositional logic. Quantifiers are things such as "and" and "or". You learn the hard way of proving things and making sentences accucrate.
@Henry Yes that is sad. I found once a red small book about something logic and it clarified me things. In my undergraduate math, it was just like that -- you need to connect the dots yourself...
I don't have the name of the book but it was fast reading and it was red, small logic book. If you have a good university lib nearby, try to find it.
<--- yes ProofWiki has very down-to-earth -explanations, they have also very student-biased attitude.
@Henry Do not hesitate to ask stupid question in Main -site. There are rigorous way of explaining simple logic things eventually, it is good if you could get some sort of broad view of things fast,.
"mathematical logic" is totally different area then what you are studying here. Start simple first.
propositional logic > Quantifier logic > predicate logic > ... > symbolic logic > ... > mahematical logic > ... > fuzzy logic > ... > perhaps even more, very broad area of research...
Google sucks if you try to search the word "logic". Do not do that, ask instead stupid questions.
(a goad reason to study mathematics is to beat google :P
...a single word with millions of different meanings...
@PeterTamaroff - If the oscillation of $f$ on $[a - \delta, a + \delta]$ is smaller than \epsilon, then it's also smaller than \epsilon on the open interval. No?
@PeterTamaroff - I can see it now. What I'm trying to say is that with the uniform continuity of f on the closed interval, you can ensure that oscillation is smaller than any epsilon on the closed and open intervals.
@PeterTamaroff Does the size of Kindle matter at all? Or is it just good that it is small? I am wondering differences between Kindle 3G paper-white and Kindle 69
Let $\Gamma = SL_2(\mathbb{Z})$.
Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$.
Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$.
Let $\alpha = \left( \begin{array}{ccc}
p & q \\
r & s \end{array} \right)$ be an element of $\Gamma$.
We write $f^\alpha(x, y) =...
