In which we meet the definition of continuity for functions.
- Definition of what it means for a complex-valued function to be continuous at a point and on a set.
- Lemma 14: Let be a subset of , and let and be functions from to . Let be a point in .
- If there is some such that for all in , then is continuous on .
- If and are continuous at , then so is the pointwise sum .
- If and are continuous at , then so is the pointwise product .
- Suppose that for all in . If is continuous at , then so is .
- Exercise: show that the function given by is continuous on . Together with Lemma 14, this shows that polynomials are continuous.
- Lemma 15: Let be a subset of . Let be a function from to that is continuous at some point in . Let be a sequence in such that as . Then as .
Understanding today’s lecture
Define the function by setting if is rational and if is irrational. At which points is continuous?
Pick some functions and check whether you can use the definition of continuity to show that they are/aren’t continuous. (We haven’t officially defined functions such as , , and in this course, so you might want to steer clear of those for now – or think about how to define them first.)
What’s the most exotic function that you can study? Can you build a function with exactly one point of continuity? Exactly two? Exactly one hundred? Infinitely many points of continuity and infinitely many points of discontinuity? Exactly one point of discontinuity? Exactly two? Can you build a function that is continuous on some intervals and not on others? You should try to get a feeling for what is possible. You could try this for real-valued functions and functions from the complex numbers to the complex numbers.
Remind yourself (without looking at your notes!) what the definition of continuity is. What’s the negation? That is, what does it mean to say that is not continuous at a point ?
If you are feeling a bit alarmed by the definition of continuity, then you might like to read Tim Gowers’s helpful remarks.
You can read about how to define continuity in more general situations (such as between metric spaces) on the Wikipedia page.
Preparation for Lecture 8
In Lemma 14 we considered various ways to combine functions to obtain another. What happens with composition of functions? If and are defined appropriately (what does that mean?) and are continuous at appropriate points (what does that mean?), is the composition continuous at an appropriate point?