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 aWe 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 aWe 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?

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 \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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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 […]

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