## 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 theoremLet $f:[a,b] \to \mathbb{R}$ 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?

### 7 Responses to “Analysis I: Lecture 8”

1. apgoucher Says:

Hey, Vicky, you should have reserved the Goldilocks story for when we do the Bear Category Theorem. ;P

2. theoremoftheweek Says:

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?

3. skhan Says:

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!

4. theoremoftheweek Says:

Wow!

5. Zhixun_Liang Says:

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.

6. Analysis I: Lecture 9 « Theorem of the week Says:

[...] Expositions of interesting mathematical results « Analysis I: Lecture 8 [...]

7. Analysis I: Lecture 17 | Theorem of the week Says:

[...] 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 [...]