- 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 theThere 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?
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?