Analysis I: Lecture 8

In which we consider continuous functions, Goldilocks, porridge, and the intermediate value theorem.

Lemma 16: Let $E$ be a subset of $\mathbb{C}$ and let $a$ be a point in $E$ . Let $f$ be a function from $E$ to $\mathbb{C}$ . Suppose that if $(z_n)_{n=1}^{\infty}$ is a sequence in $E$ such that $z_n \to a$ as $n \to \infty$ , then $f(z_n) \to f(a)$ as $n \to \infty$ . Then $f$ is continuous at $a$ . We proved this by contradiction: we supposed that $f$ is not continuous at $a$ and built a sequence $(z_n)_{n=1}^{\infty}$ such that $z_n \to a$ as $n \to \infty$ but $f(z_n) \not\to f(a)$ as $n \to \infty$ .
Lemma 17: Let $U$ and $V$ be subsets of $\mathbb{C}$ . Let $f:U \to \mathbb{C}$ and $g:V \to \mathbb{C}$ be functions such that $f(z) \in V$ for all $z \in U$ . Suppose that $f$ is continuous at some $a \in U$ and that $g$ is continuous at $f(a)$ . Then the composition $g \circ f$ is continuous at $a$ . We proved this using Lemmas 15 and 16. Exercise: prove it directly from the definition of continuity.
Theorem 18: (Intermediate value theorem) Let $f:[a,b] \to \mathbb{R}$ be a function that is continuous on $[a,b]$ with $f(a) < 0 < f(b)$ . Then there is some $c$ in $(a,b)$ such that $f(c) = 0$ . We proved this using ‘zooming in’ or ‘interval bisection’ or ‘lion hunting’ — this was the strategy that we used in lectures to prove the Bolzano-Weierstrass theorem.

Understanding today’s lecture

Can you prove Lemma 17 directly from the definition of continuity? This is a good exercise for developing familiarity with the definition.
Give another proof of the intermediate value theorem along the following lines. Let $S = \{x \in [a,b]: f(x) < 0\}$ . Show that this set has a supremum, say $s$ . Show that $a < s < b$ , and that $f(s) = 0$ . (~~I’ll put a proofsorter activity on the~~ There is now a proofsorter activity on the course webpage ~~later this week~~, but I recommend trying to prove it yourself first.)
Show that every real polynomial of odd degree has at least one real root. Is the same true for polynomials of even degree?

Preparation for Lecture 9

Here are some functions defined on subsets of the real line $\mathbb{R}$ . For each function $f:I \to \mathbb{R}$ , decide whether $f$ is bounded, and if so whether $f$ attains its bounds. (E.g. if $f(x) \leq 100$ for all $x$ , is there some value of $x$ such that $f(x) = 100$ ?

$f : (0,1) \to \mathbb{R}$ , $f(x) = 1/x$ .
$f: (1,2) \to \mathbb{R}$ , $f(x) = 1/x$ .
$f: [1,2] \to \mathbb{R}$ , $f(x) = 1/x$ .
$f : [-1,1] \to \mathbb{R}$ , $f(x) = 1/x$ if $x \neq 0$ and $f(0) = 1$ .

Can you generalise your findings? Can you find any necessary or sufficient conditions for $f : I \to \mathbb{R}$ to be bounded and to attain its bounds?

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This entry was posted on February 4, 2013 at 12:19 pm and is filed under Cambridge Maths Tripos, IA Analysis I, Lecture. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

7 Responses to “Analysis I: Lecture 8”

apgoucher Says:
February 4, 2013 at 5:53 pm
Hey, Vicky, you should have reserved the Goldilocks story for when we do the Bear Category Theorem. ;P
theoremoftheweek Says:
February 4, 2013 at 5:57 pm
Thanks for that! Here’s a link for those puzzled by the remark:

http://en.wikipedia.org/wiki/Baire_category_theorem

Who knew that so much of mathematics was related to fairy tales? Any more examples, anyone?
skhan Says:
February 4, 2013 at 7:21 pm
It seems someone managed to link the Bolzano-Weierstrass Theorem with Red Riding Hood!

http://people.maths.ox.ac.uk/macdonald/errh/101_analysis_bedtime_stories_%28epsilon_red_riding_hood%29.pdf

Perhaps we should add our own story to the book!
theoremoftheweek Says:
February 4, 2013 at 9:02 pm
Wow!
Zhixun_Liang Says:
February 5, 2013 at 6:12 am
I recommend a very good book called “A Companion to Analysis” by T. W. Korner. He makes analysis a very easy subject. Suitable for Analysis 1 and 2.
Analysis I: Lecture 9 « Theorem of the week Says:
February 6, 2013 at 12:43 pm
[...] Expositions of interesting mathematical results « Analysis I: Lecture 8 [...]
Analysis I: Lecture 17 | Theorem of the week Says:
February 25, 2013 at 12:27 pm
[...] function from to . We then carefully checked the remaining properties using these. We used the intermediate value theorem for (vi). We noted that (ii) and (vi) show that the function is a group isomorphism between the [...]

Theorem of the week