Review for Examination 1

General Rules for Arguments about Continuity:

To show a specific function is continuous; If there is a metric use it in an argument.
To show a general property of continuous functions; Use an Open Sets/Boolean Algebra argument, often even if it is about Metric Space.

Consider problem 2.7, was given by the formula

You were asked to show it was continuous by an argument, and then by showing that the preimage of every set was open.

Given that you calculated a for every completing part one. How does the rest of the argument go?

Let be an open set. We need to show the is open. Let Since is open we can find some such that

. By the argument we know that we can find such that

Hence .

In particular, we proved part 2 by refering back to the proof in part 1.

Consider problem 3.4, was such that the preimage of every closed set was closed. You were asked to show that $\QTR{Large}{f\ }$ was continuous.

The proof involves some set-theoretic formulii from the "Background" notes.

If $\QTR{Large}{A}$ and $\QTR{Large}{B}$ are sets, there is some algebra associated with maps ( functions) from $\QTR{Large}{A}$ to $\QTR{Large}{B.}$

Let $\QTR{Large}{B\ }$ be two subsets of $\QTR{Large}{B}$ then

$\qquad$ 1.

$\qquad$ 2.

In particular, if such that then

such that

Thus if is closed , is open.

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To Show Sets are Closed:

Really, the only tool we have at the moment is to show that the complement is open. Remember, showing a set is not open does not mean that it is closed.

Problem 2.3 asked you to show that was closed for any number $\QTR{Large}{x}$ . To do this you need to show is open.

Thus for any you need to find a such that Anything will do.