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
was
continuous.
The proof involves some set-theoretic formulii from the "Background" notes.
If
and
are sets, there is some algebra associated with maps (
functions) from
to
Let
be
two subsets of
then
1.
2.
In particular, if
such that
then
such that
Thus if
is
closed
,
is open.
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
.
To do this you need to show
is
open.
Thus for any
you need to find a
such
that
Anything
will do.