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# Class Contents

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Properties of Sequentially Compact Sets

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Continuous functions are bounded on sequentially compact sets

Sequentially compact sets are important because continuous functions
defined on sequentially compact sets have some very useful properties,
which they do not have in general when defined on non-compact sets. These
include being bounded, and having points at which they reach their
maximum and minimum values. In this section we shall prove most of these
properties, which, in the main, follow quite quickly from the definition of
sequential compactness. We can rely on the results of the last section
to tell us that sequentially compact sets are very common, and to help us
recognize them in practice.###
Invertible continuous functions on sequentially compact sets

Suppose that `f` is a continuous function from `X` to
`Y`, which is one-to-one and onto (also called *bijective*).
Then `f` has an inverse function from `Y` to `X`.
It seems very reasonable to expect that since `f` is continuous
then its inverse will also be continuous. After all, we are thinking
that `y=f(x)` is going to vary continuously as `x` varies,
so surely, `x` will vary continuously as `x` varies.###
Continuous functions attain their bounds on sequentially compact sets

In
Corollary 6.8
we saw that a continuous function defined on a sequentially compact set is always bounded. In fact, something much stronger is true: A continuous function defined on a sequentially compact set has a point at which it *attains* its greatest value. Thus, not only does `sup(f(X))` exist, but there is an `x`_{0} at which `f(x`_{0})=sup(f(X)).

Now, if `A` is closed and bounded in Euclidean space then by
Theorem 6.5, it is sequentially compact. Theorem 6.6 says that any
continuous image of `A` must be sequentially compact, and so by
Proposition 6.4, the image is also closed and bounded. In general the
continuous image of closed sets isn't closed, but this example shows
that quite often continuous functions on Euclidean space (in particular,
on the real line) will map closed set to closed sets. We summarize this
fact in a corollary:

In general, if a continuous function is defined on a sequentially compact set then its range will be sequentially compact. Thus, since the range will be a bounded set (again by Proposition 6.4), the function will be bounded:

Example:

We shall study `C[a,b]` in depth in subsequent chapters, and
shall even work an extended example with it later in this chapter.

It is a very surprising fact, that the definition of continuity alone,
does not ensure that the inverse of `f` will necessarily be
continuous! Fortunately, in most of the situations that interest us, we can show that the inverse of a continuous
function *will* still be continuous. But to do that, we need an
extra ingredient, which is usually provided by compactness.

Because it seems implausible that there is a continuous invertible map which does
not have a continuous inverse, we'll start with an
example to show that this really can happen. However, although an
example is convincing, it doesn't necessarily show *why* this
unexpected turn of events is coming about.

So after the example, we'll see what happens if one sets out to prove that the inverse of a continuous function must be continuous, using only the definition of continuity. The logical problems that we run into when we try this illustrate why, after all, it is not so surprising that the inverse of a continuous function need not be continuous.

Example:

Remark:

The extra hypothesis that we need in order to show that the inverse is
also continuous, is that `f` be defined on a sequentially compact
set. Proposition 6.10 will prove that, but first we need a lemma, which
is also useful in its own right for recognizing sequentially compact sets.

As an application of this result, we can give a quick proof of the fact that the logarithmic functions are all continuous.

Example:

Put another way, this says that when a continuous function is defined on a sequentially compact set, it has a *maximum*. Not surprisingly this fact has important implications for studying differentiable functions, where maxima and minima of functions play an important role. In the present section we shall prove this result, and in the next, we shall apply it to prove the Mean Value Theorem, which is the most important tool for understanding real differentiable functions.

Of course, by applying the proposition just proved to `-f` in place of `f`
we immediately obtain the corresponding result for minima:

Continued in the next class...