## Analysis I: Lecture 9

In which we study continuous functions further, looking at boundedness and inverses.

• Theorem 19 (A continuous function on a closed, bounded interval is bounded and attains its bounds)  Let $f:[a,b] \to \mathbb{R}$ be a function that is continuous on the closed, bounded interval [a,b].  Then there is some $K$ such that $|f(x)| \leq K$ for all $x \in [a,b]$ (“f is bounded”).  Moreover, there are $y_1$ and $y_2$ in $[a,b]$ such that $f(y_1) \leq f(x) \leq f(y_2)$ for all $x \in [a,b]$ (“f attains its bounds”).  We proved this in two parts, using Bolzano-Weierstrass for each.
• Definition of what it means for a function to be increasing, strictly increasing, decreasing, strictly decreasing, or monotone.
• Theorem 20: Let $f:[a,b] \to \mathbb{R}$ be continuous and strictly increasing on $[a,b]$.  Let $c = f(a)$ and $d = f(b)$.  Then $f:[a,b] \to [c,d]$ is a bijection, and the inverse $g:[c,d] \to [a,b]$ is also continuous and strictly increasing.  We proved this in several pieces.  To show that $f$ is surjective, we used the Intermediate value theorem.
• I handed out the second examples sheet.

#### Understanding today’s lecture

• Give another proof that $f$ in Theorem 19 must attain its bounds, along the following lines.  Let $K = \sup\{f(x) : x \in [a,b]\}$ and suppose that $f(x) < K$ for all $x \in [a,b]$.  Define $g : [a,b] \to \mathbb{R}$ by $g(x) = 1/(K-f(x))$.  Show that $g$ is a continuous function on a closed, bounded interval, and obtain a contradiction by showing that $g$ is not bounded.
• Can you ‘break’ Theorem 19?  That is, can you find examples to show that each of the conditions is necessary?  This is a really good way to understand the result.
• Are there any functions that are both increasing and decreasing?  Both strictly increasing and strictly decreasing?
• Play around with some examples of continuous functions that are strictly increasing to get a feel for Theorem 20.  Do we need the function to be continuous and/or strictly increasing for the result to be true?

The Tricki has an example discussing the use of the Bolzano-Weierstrass theorem to show that a continuous function on a closed, bounded interval must be bounded.  It also has a piece on how to use the fact that a continuous function on a closed, bounded interval attains its bounds.  (Don’t worry about the references to compact sets: in the reals these are just closed, bounded intervals.  You’ll learn more about this in Metric and Topological Spaces next term.)

Tim Gowers has written an interesting piece on how one might think of a proof that a continuous function on $[0,1]$ is bounded.

I have just come across an intriguing book by David Bressoud, called A Radical Approach to Real Analysis.  There’s a copy in the Moore Library.  It offers a different perspective on the material, looking at how the ideas that we are meeting in the course were studied historically and using that as the basis of an introduction to the subject.  It looks as though it has rather friendly descriptions of some difficult ideas: it might be worth a look if you would like a different style of exposition.

#### Preparation for Lecture 10

When is it possible to differentiate a (real-valued) function?  Here are some examples of functions $f : \mathbb{R} \to \mathbb{R}$; try your own too.  Are they differentiable everywhere, or at just some points, or nowhere?

• $f(x) = x^2$.
• $f(x) = |x|$.
• $f(x) = 1$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational.

Try to formulate a precise definition of what it means for a function $f: \mathbb{R} \to \mathbb{R}$ to be differentiable at a point $x \in \mathbb{R}$.  Phrase your definition using limits, and then using $\epsilon$ and $\delta$.

Is every continuous function differentiable?  Is every differentiable function continuous?  Does every differentiable function have a continuous derivative?