*In which we meet some really important results about differentiable real-valued functions.*

- Theorem 22 (Chain rule)
*Let and be subsets of . Let be a function such that for all , and let be a function. If is differentiable at some and is differentiable at , then is differentiable at and .*We proved this by picking a convenient way of expressing differentiability and then combining the resulting expressions for and . - Theorem 23 (Rolle‘s theorem)
*Let be a function that is continuous on and differentiable on . Suppose that . Then there is some such that .*To prove this, we used the fact that a continuous function on a closed, bounded interval is bounded and attains its bounds (Theorem 19). We used this to pick a point where attains its maximum and checked that this point did the job. - Theorem 24 (Mean value theorem)
*Let be a function that is continuous on and differentiable on . Then there is some such that .*We proved this by ’tilting’ Rolle’s theorem: we applied Rolle to the auxiliary function defined by . - Corolla
*ry 25 Let be continuous on and differentiable on .**If for all , then is strictly increasing on .**If for all , then is increasing on .*- (Constant value theorem)
*If for all , then is constant on .*

With the help of the mean value theorem, this was pretty straightforward. But it’s a surprisingly subtle result!

#### Understanding today’s lecture

To help get a feeling for what these theorems say, and how their proofs work, you could try running the arguments on some examples.

Do these theorems hold for functions from subsets of to ? If not, where have we used special properties of the real numbers?

What other questions might you ask yourself? Please do share suggestions in the comments below.

#### Further reading

Tim Gowers has written a piece called “What is the point of the mean value theorem?“, so if you can’t answer that question then you might like to read his thoughts on the matter.

If you (very naturally) wonder where all this real analysis might lead to, then I recommend *The Princeton Companion to Mathematics*. This is a very large book, but you should dip into it and read the bits that appeal to you. It’s ideal for undergraduates who want to get a feeling for how their current studies fit into the larger mathematical edifice.

#### Preparation for Lecture 12

- We thought a little about inverse functions in Theorem 20. If we have a function that satisfies the conditions of that theorem and that satisfies the extra condition of being differentiable, can we differentiate its inverse? If so, what is its derivative?
- Polynomials are nice friendly functions: they are, in many ways, nicely behaved and easy to understand. It would be handy to be able to approximate non-polynomials by polynomials. How might we do that? (Remember that this is analysis, so if we have ‘error terms’ then we have to be very precise about them!)

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