Monday, January 21, 2013

More on monotonic sequences and a brief introduction to cardinality

Theorem: If $I$ is an interval, $f: I \rightarrow \mathbb R$ be monotone increasing, and $c\in I$ is not an end point of $I$, then
(i) $\lim_{x\to c^+} f(x) = \sup\{f(x) : x\in I, x > c\}$, and
(ii) $\lim_{x\to c^-} f(x) = \inf\{f(x) : x\in I, x < c \}$.

Corollary: Let $I$ be an interval and $f : I \to \mathbb R$ be monotone increasing, and $c\in I$ not an end point of $I$. TFAE
(i) $f$ is continuous at $c$
(ii) $\lim_{x\to c^+}f(x) = \lim_{x\to c^-} f(x)$
(iii) $\sup\{f(x) : x \in I, x<c\} = f(c) = \inf\{f(x) : x\in I, x>c\}$

Proof: Follows immediately from previous Theorem. Today was a slow day in MATH255.

Definition: If $I$ is an interval, $f : I\to \mathbb R$ is increasing on $I$, and $c\in I$ is not a you-know-what, we define the jump of $f$ at $c$ as
$J_f(c) = \inf\{f(x) : x\in I, x>c\} - \sup\{f(x) : x \in I, x<c\}$, or equivalently
$J_f(c) = \lim_{x\to c^+}f(x)-\lim_{x\to c^-} f(x)$

For end points of an interval $[a, b]$, $J_f(a) =\lim_{x\to c^+}f(x) - f(a)$.

Corollary: If $I$ is an interval, and $f$ increasing on $I$, then $f$ is continuous at a point $c\in I$ iff $J_f(c) = 0$

Class then diverged in to a brief discussion of countable sets, which seems like it will eventually lead to a discussion of why monotonic sequences are continuous "almost everywhere" (Trudeau's words). I could type out a bunch of stuff about that here, but it if you're not comfortable with the idea it would probably be better for you to read about countability in the text (I think they call it denumerable instead of countable though. Something like that). Other than that you could probably read some Wikipedia articles about Hilbert's Hotel, the diagonal argument for the countability of the rationals from the countability of the Natural numbers, as well as Cantor's theorem concerning power sets and Russel's Paradox. GLHF