In which we learn what it means to say that a function is differentiable, and prove some basic properties of differentiation.
- Definition of the limit of a function.
- Definition of differentiability, and the derivative.
- Lemma 21: Let be a subset of , and let and be functions from to . Let be a point in .
- If is differentiable at , then is continuous at .
- If there is some such that for all , then is differentiable at with .
- If and are differentiable at , then so is their sum , and .
- ‘Product rule’ If and are differentiable at , then so is their product , and .
- If is differentiable at and for all , then is differentiable at , and .
We noted that (iv) and (v) together give the quotient rule: . We proved the five statements using the definition of differentiability and basic facts about limits of functions.
Understanding today’s lecture
- Suppose that . Show that is continuous at if and only if as .
- Formulate and prove an analogue of Lemma 2 for limits of functions.
- Check that you understand the definition of differentiability by showing directly that some functions are continuous and others are not.
- Start your ‘functions grid’. The columns correspond to examples of functions, and the rows to properties that a function may or may not have (e.g. continuous at one point, continuous on the domain, bounded, increasing, strictly decreasing, etc.). Put a tick if the function has the property, and a cross if it does not. Make sure that you can carefully justify each tick and cross! The aim is to explore the extent of each property by considering examples, so you should make sure you have some examples that are ‘typical’ and some that are ‘extreme’. You should also try to see how the properties relate to each other: does one property imply another?
Some of you may be interested in Tim Gowers’s thoughts on how to approach analysis problems in an automatic sort of way. There’s a Tricki article in a similar vein. This sort of approach is sometimes described as ‘syntactic’, as opposed to ‘semantic’ approaches where one develops an intuition and then tries to turn it into a formal argument. I suspect that some people find that one of these approaches comes more naturally to them than the other, but it’s important to be comfortable with both styles since sometimes one is much more convenient than the other.
There’s now a proofsorter exercise for the Intermediate value theorem on the course webpage.
I briefly mentioned compactness in the last lecture; you’ll learn about this in the course on Metric and Topological Spaces next term. There’s a nice article by Terry Tao that discusses what compactness means.
Preparation for Lecture 11
- What does the chain rule say? Can you prove it rigorously?
- Take a differentiable function that has . What can you say about the gradient of in between and ? Can you prove a theorem?
- If a differentiable function is increasing, does it have non-negative gradient? If a differentiable function has non-negative gradient, must it be increasing? What happens if we replace “increasing” by “strictly increasing”? If a differentiable function is constant, does it have zero derivative? If a differentiable function has zero derivative, must it be constant? Try to justify all your assertions (proof or counterexample).