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# Class Contents

##
Uniform Continuity

We'll start by re-examining the definition of continuity for a function between two metric spaces.

Of course this rather begs the question, so recall also the definition of continuity at a point:

Now we pose the question: What does `delta` depend on?

Clearly we expect that the smaller `epsilon` is, the smaller we will have to take `delta`. But since `x _{0}` was set before we started to look for

Example:

Remarkably enough, oftentimes for functions that crop up in practice it is possible to
find values of `delta` which work for a given `epsilon` and for *all*
`x _{0}`. When this happens we say that the function is

The reason that this remarkable, and frequently useful, property is fairly common, is tied up in sequential compactness. We start with a formal definition of uniform continuity.

Remark:

Example:

There is a simple graphical interpretation of uniform continuity. We saw in
an
interactive demo
that a function was continuous at a point `x _{0}`
if, given a fixed height

By the same token, we can say that `f` is uniformly continuous
if, given a diameter of `epsilon` we can always cut a tube of
diameter `epsilon` to short enough a length (namely, `delta`) that
it can run freely along the curve withough bumping into the sides of the
tube. The following picture illustrates this idea:

From this point of view, it is clear that one thing which could cause a
continuous function to fail to be *uniformly* continuous would be
is the slope of the line becomes too large. The following theorem shows
that, conversely, if the slope is never too large, the function must be
uniformly continuous.

Remark:

This last result relies on differentiability, and is only applicable to real-valued functions on the real line. The following result shows why uniform continuity is in fact a very common property, despite being, apparently, much stronger than simple continuity. Not surprisingly, the key ingredient is sequential compactness.

Example:

The function `f(x)=x^{1/2}` is uniformly continuous on
`(0,1)`, even though it has an unbounded derivative on that interval.

In the next section we shall introduce the definition of
*compactness* (as opposed to * sequential compactness*), and
show that the two definitions are, in fact, equivalent.