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

- Lemma 16:
*Let be a subset of and let be a point in . Let be a function from to . Suppose that if is a sequence in such that as , then as . Then is continuous at .*We proved this by contradiction: we supposed that is not continuous at and built a sequence such that as but as . - Lemma 17:
*Let and be subsets of . Let and be functions such that for all . Suppose that is continuous at some and that is continuous at . Then the composition is continuous at .*We proved this using Lemmas 15 and 16. Exercise: prove it directly from the definition of continuity. - Theorem 18: (Intermediate value theorem)
*Let be a function that is continuous on with . Then there is some in such that .*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 . Show that this set has a supremum, say . Show that , and that . (
~~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?

#### Further reading

There’s a link to a blog post about the intermediate value theorem in the summary of today’s lecture. There are lots of books and websites where you can read about this theorem — please leave a comment below if you have any recommendations. There’s a link above to a recipe for porridge.

#### Preparation for Lecture 9

Here are some functions defined on subsets of the real line . For each function , decide whether is bounded, and if so whether attains its bounds. (E.g. if for all , is there some value of such that ?

- , .
- , .
- , .
- , if and .

Can you generalise your findings? Can you find any necessary or sufficient conditions for to be bounded and to attain its bounds?

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

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?

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!

February 4, 2013 at 9:02 pm

Wow!

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.

February 6, 2013 at 12:43 pm

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

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 […]