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PROFESSOR: Hi.
Our lecture today actually has us backtrack a little bit to a
concept which is actually more fundamental than that of
differentiability.
It's a topic which is called 'continuous functions'.
In other words, our lecture today is concerned with a
topic called Continuity.
And actually, the topic of continuity had its roots way
back at the beginning of our course when we first raised
the question: does the limit of 'f of x' as 'x' approaches
'a' equal 'f of a'?
Remember, when we first started talking about the
limit concept, our intuitive approach was to say, look-it,
as 'x' gets arbitrarily close to 'a', 'f of x' gets
arbitrarily close to 'f of a'.
And we saw that this particular definition had a
few loopholes in it, even though intuitively this is
what we would have liked.
And so what I want to do today is to see what happens if we
can be sure that the limit of 'f of x' as 'x' approaches 'a'
equals 'f of a'.
What is implied if this happens
to be a true statement?
And my first claim is that it is implicitly implied that 'f
of a' must at least make sense.
Otherwise, we wouldn't have written it over here.
In other words, 'f' must be defined when 'x' equals 'a'.
'f of a' must be defined.
Now to correlate that with material that we've had
previously, let's go back to an example that we've
discussed before, or at least an example which is close to
something we've studied before.
Let's define 'f of x' to be 'x squared minus 1' over 'x - 1'.
Now you see as long as 'x' is not equal to 1, we can cancel
'x - 1' from both numerator and denominator, leaving this
equal to 'x + 1'.
And we then see what?
That the limit of 'f of x' as 'x' approaches
1 is equal to 2.
On the other hand, notice that 'f of 1' turns out to be 0/0
in this case, which is undefined.
In other words, in this particular case, the limit of
'f of x' as 'x' approaches 1 is not equal to 'f of 1' if
only because 'f of 1' isn't even defined.
And again, by way of a review, let's look at this thing
pictorially.
Remember, 'x squared minus 1' over 'x - 1' is 'x + 1' except
when 'x' equals 1.
Consequently, to graph 'y' equals 'x squared minus 1'
over 'x - 1', we simply graph 'y' equals 'x + 1' with the
point corresponding to 'x' equals 1, meaning the point 1
comma 2 deleted.
In other words, this point deleted is the graph 'y'
equals 'x squared minus 1' over 'x - 1'.
You see again there is no definition at 'x' equals 1
because this point is missing.
It would be the 0/0 form.
Now, to be sure, we can clean this up in the sense that we
could define a new function 'g of x' to be 'x squared minus
1' over 'x - 1' as long as 'x' is not equal to 1 and define
it to be 2 when 'x' equals 1, which, by the way,
geometrically is just a fancy way of writing the equation of
the line 'y' equals 'x + 1'.
In other words, all we're saying is that letting 'g of
x' be 2 when 'x' equals 1 plugs the little hole to the
periods over here.
Now again, I hope that this seems fairly familiar from our
discussion on limits, but for the time being, all I want us
to see is that as soon as you write down that the limit of
'f of x' as 'x' approaches 'a' equals 'f of a', you at least
imply that 'f of a' must be defined.
Let's see another property that's implied by our
definition.
I claim, as you might expect, that if the limit of 'f of x'
as 'x' approaches 'a' is 'f of a', that means that 'f of x'
is near 'f of a' when 'x' is near 'a'.
In other words, my claim is that the curve 'y' equals 'f
of x' is unbroken in a neighborhood
of 'x' equals 'a'.
And to see what I mean by that, let's look at this
picture over here.
Suppose, for example, that our curve 'y' equals 'f of x'--
let me darken this up so that we can see it a little better.
Suppose that curve had a break at 'x' equals 'a', OK?
Now, let's say 'f of a' is over here.
What I could do is now pick an interval surrounding 'f of a',
if there was a break over here, such that this band
never touched or included the top curve here.
Now notice, from this point of view, that no matter how close
'x' gets to 'a', as long as 'x' is greater than 'a', 'f of
x', which is up here on this dark curve, can never be
within epsilon of 'f of a'.
In other words, if there's a break in the curve I could
always fix it up so that the limit of 'f of x' as 'x'
approaches 'a' is not equal to 'f of a'.
At any rate, with these two properties as motivation, let
us now give our basic definition.
And the basic definition is simply this: a function 'f' is
called continuous that 'x' equals 'a' precisely if the
property that we were discussing is present, namely,
if the limit of 'f of x' as 'x' approaches 'a' is
equal to 'f of a'.
We generalize this definition and say 'f' is called
continuous on the entire interval 'I', or on the
interval 'I', if the limit of 'f of x' as 'x' approaches 'a'
is equal to 'f of a' for each 'a' in 'I'.
In other words, notice that our first definition is a
local property.
Namely, we define continuous or
continuity at a given value.
And then if the function happens to be continuous at
every point, then we call the function itself continuous.
What the thing means pictorially is that if the
function is continuous at 'x' equals 'a', it means that in
terms of the graph of the function, that in a
neighborhood of the point 'a' comma 'f' of a on the graph,
in a neighborhood of that point, in other words, in a
sufficiently small interval surrounding that point, the
curve itself must be unbroken.
Now, you see, this may sound trivial.
Remember, when we first started talking about limits,
we had the feeling that limit of 'f of x' as 'x' approaches
a should always equal 'f of a'.
And the reason for this was that instinctively we seem to
always think of continuous functions.
We think of what?
Things changing in such a way the graph of the change is
unbroken even though, of course, there are places where
discontinuities occur.
At any rate, what I'm saying is that many of the results
that I now want to discuss with you, which are very
important properties of continuous functions, may seem
self-evident because we keep thinking about any function as
being continuous.
But I will try to emphasize the fact that these properties
are not true if the function is not continuous.
I also would like to show a balance between geometric
ideas and analytic ideas.
In other words, let me start off by seeing what things seem
to follow about continuous functions based primarily on
the graph idea.
For example, I claimed that continuous functions assume
their maximum and minimum values on any closed interval.
Now, this sounds like a big mouthful, and it also sounds
kind of trivial.
You say doesn't any function have a maximum and minimum
value on a closed interval?
And the answer is no.
For example, take the curve 'y' equals '1
over ''x - 1' squared''.
Notice that when 'x' is 1, 'y' is infinite, which means that
if we graph this particular function, we find that the
graph does something like this.
In other words, it goes--
it jumps up here, comes back, et cetera.
Now, what is the highest value or what is the highest point
on our graph here?
Well, notice that the graph is broken here.
It's discontinuous when 'x' equals 1.
Notice that the curve rises to infinity.
In other words, the maximum value is undefined on this
particular closed interval.
You see there's a jump over here.
Of course, the fact that this is an infinite jump may make
you feel uneasy if you don't feel
comfortable with infinity.
So let me paraphrase this just a little bit in terms of a
finite jump.
Let me write down the following function.
Let's think of the function 'f of x' equals 'x' if 'x' is
less than 1.
Then as soon as 'x' is at least as big as 1, the
function becomes 'minus x'.
In other words, graphically, we have the line 'y' equals
'x' from 0 up to 1, but not including 1.
And then at 1, the function jumps down to minus 1 and
becomes the line 'y' equals 'minus x'.
Let's look, for example, at the closed interval
again from 0 to 2.
What is the biggest value?
What is the maximum that the function can have if 'x' is
between 0 and 2?
Well, notice that as 'x' gets closer and closer to 1, 'f of
x' gets closer and closer to 1, But 'f of x' never equals
exactly 1, because when 'x' is 1, the curve
jumps down to here.
In other words, notice that as 'x' gets arbitrarily close to
1, 'f of x' increases and gets arbitrarily close to but
never equals 1.
In other words, notice that by picking 'x' to be less than 1,
we can make 'f of x' as close to 1 as we want, but we can
never make it exactly equal to 1 because of this jump which
takes place over here.
You see, this discontinuity causes us a little bit of
difficulty.
By the way, we can use this example from
another point of view.
If we just look at the part where it says 'f of x' is
equal to 'x' where 'x' is less than 1, that means that we'll
just look at this part of the curve.
Notice that this part of the curve is unbroken.
And yet as 'x' approaches 1, 'f of x' approaches but never
equals 1, which means that to get the maximum value in here,
I would have to include the end point 'x' equals 1.
In other words, notice that even if the function is
continuous, if the interval is open, it may not assume the
maximum value.
In other words, you may not be able to find the value of 'x'
in that interval such as the function will be maximum at
that particular value.
Now again, this is making a mountain out of a mole hill in
terms of your intuition because these are result which
I'm sure you believe are true anyway.
In other words, it seems to be true.
But all I hope is that these little examples here show you
what the importance of continuity is, because as we
progress later in our work, these so called fine points
are going to become crucial in many of our proofs.
And by the way, I think it's a truism to say that if you make
fine points seem very, very important before you get to
use them and don't emphasize them, people tend to think
that you overestimated your case.
So I prefer not to beat this to death and wait until such
times in our course that we need these results before I
emphasize them more.
So these are results which are in the textbook, which I'll
test you on in the exercises and the like, but just want to
go through with you now so that you get an overall idea.
Another result that's rather clear geometrically is
something called the intermediate value theorem.
Suppose that 'f' is continuous on the closed interval from
'a' to 'b'.
And without loss of generality, let's suppose 'f
of a' is less than 'f of b'.
The same kind of an argument would hold if 'f of b' were
less than 'f of a', but I just need some sort of an
orientation.
Now I say let 'm' be any number such that m is what?
Between 'f of a' and 'f of b'.
In other words, 'm' is greater than 'f of a' but
less than 'f of b'.
You see, if 'f of b' were less than 'f of a', I would just
reverse the inequality signs here.
At any rate, what my claim is that we can find the number
'c' in the open interval from 'a' to 'b' such that 'f of c'
equals 'm'.
Again, this may sound kind of complicated.
All it says in terms of a picture is this.
If our curve goes from point 1 to point 2 where the height of
'p1' is 'f of a' and the height of 'p2' is 'f of b',
then it must assume every intermediary height between 'f
of a' and 'f of b' on that particular interval.
Again, you can think of that in terms of an
automobile, if you want.
If an automobile goes from a speed of 20 miles an hour to a
speed of 30 miles per hour in what we think of as being
continuous motion, then at some time in that interval, it
must've had any particular speed you want to mention
between 20 and 30 miles.
In other words, we don't visualize a car going from 20
miles an hour to 30 miles an hour, say, without at one time
having passed the speed of 27 miles per hour
or some such thing.
And geometrically, you see, this proof is very simple,
namely, draw the line 'y' equals 'm'.
And now what you're saying is you want to get from this
point to this point.
And the idea is that the only way you can get from this
point to this point without crossing this line is if
someplace along the way, you jump over this line.
In other words, geometrically, all you're saying is lookit,
if I have to join 'p1' and 'p2' with an unbroken line,
then I must cross the line 'y' equals 'm' someplace.
By the way, notice I say that at least one place, the curve
could have done something like this, for example.
But the important point is what?
There's at least one number 'c' such that 'f
of c' equals 'm'.
Again, somebody might think in terms of an end run and say
couldn't we have done something like this and not
cross this line?
That brings us back again to our concept of
single-valuedness.
In other words, notice, if we remove the restriction that 'f
be a' single-valued function, in other words, if 'f' can
double back, notice that if we think of the line 'y' equals
'm' as being endless, the point remains what?
That you must someplace cross this line in going
from 'p1' to 'p2'.
But if we remove the single-valued restriction,
then the point at which you cross this line
would not have to be--
see, in this case, 'c' would not be in the open interval
from 'a' to 'b' since 'f' is not single- valued.
And what I'm hoping this discussion does for us is
gives us a good geometric feeling as to what is
happening in terms of a continuous function.
By the same token, remember our basic definition of
continuity goes back to our concept of limit.
And our concept of limit has been cemented down fairly
firmly from an analytical point of view, and that means
that we can also see these things theoretically as well
as pictorially.
And hopefully what we will do is combine pictorial and
analytic aspects to the best possible
advantage for solving problems.
But to show you what I mean by this, let's look at a few
analytical properties of continuous functions.
For example, suppose 'f' and 'g' are both continuous at 'x'
equals 'a'.
And suppose we define 'h' to be the sum of 'f' and 'g'.
In other words, 'h of x' is 'f of x' plus 'g of x', OK?
Let's compute the limit of 'h of x' as 'x' approaches 'a'.
By definition, the limit of 'h of x' as 'x' approaches 'a' is
the limit of the quantity 'f of x' plus 'g of x' as 'x'
approaches 'a'.
Now, if we look at this particular expression, notice
that because of our theorems on limits, we can do an awful
lot with this without having to make any recourse to our
picture at all.
For example, notice that we can say right away that since
the limit of a sum is the sum of the limits, this is the
limit of 'f of x' as 'x' approaches 'a' plus the limit
of 'g of x' as 'x' approaches 'a'.
Secondly, since 'f' and 'g' are continuous at 'x' equals
'a', by definition of continuous that 'x' equals
'a', this says that the limit of 'f of x' as 'x' approaches
'a' is 'f of a'.
The limit of 'g of x' as 'x' approaches 'a' is 'g of a'.
Therefore, this expression is 'f of a' plus 'g of a'.
But by the definition of 'h', this is just 'h of a'.
And if we now look at this, we see what?
That the limit of 'h of x' as 'x' approaches 'a'
equals 'h of a'.
Notice that this result came about under the assumption
that 'f' and 'g' were both continuous that 'x' equals
'a', and what have we proven over here?
This is precisely the statement that 'h' is
continuous at 'x' equals 'a'.
In other words, what we've now proven analytically is that a
sum of two continuous functions is
a continuous function.
By induction, we could prove this for the sum
of more than two.
We can in a similar way by using limit theorems prove
things like the product of two continuous functions is
continuous, et cetera.
We also were on the verge of a topic like this when we talked
about differentiability sometime back.
Namely, I claim that there is an interesting connection
between differentiable and continuous, and that the
connection is that any differentiable function is
continuous.
And by the way, the proofs utilizes that which we used in
our first lecture on derivatives in this block.
Namely, we take the expression 'f' of--
we want to show that if 'f prime of a' exists, that 'f'
must be continuous at 'x' equals 'a'.
That means we want to show that the limit of 'f of x' as
'x' approaches 'a' is 'f of a', and that's the same as
saying we want to show that this difference approaches 0
and the limit.
And the trick is we take 'f of x' minus 'f of a' and write it
as ''f of x' minus 'f of a' divided by 'x - a'' times 'x -
a', the idea being that we'll now take the limit as 'x'
approaches 'a'.
The limit of a product is the product of the limits.
The limit of this as 'x' approaches 'a' is just going
to be 'f prime of a'.
That's our definition of derivative.
And the limit of 'x - a' as 'x' approaches 'a' is just 0.
In other words, putting these steps together,
look what we get.
The limit as 'x' approaches 'a', 'f of x' minus 'f of a',
is the limit as 'x' approaches 'a' of this bracketed
expression ''f of x' minus 'f of a' over 'x - a'' times the
limit of 'x - a' as 'x' approaches 'a'.
By definition, 'f' being differentiable at 'a' means
that this limit exists and is, in fact, 'f prime of a'.
But any finite number times 0 is still 0.
And therefore, it follows that the limit of 'f of x' as 'x'
approaches 'a' is 'f of a'.
There's a legitimate analytic proof that differentiability
implies continuity.
Now, it's not necessarily true that continuity implies
differentiability.
And you see even though you can prove these things
analytically, if you don't form some picture in your
mind, you frequently will my wind up memorizing results
rather than having a feeling for them.
What I mean by having pictures and proofs coexist is this.
Instead of saying to yourself, let's see, continuity is a
weaker condition than differentiability, it's a
strong-- which one is it?
You say look-it.
We identified continuity with unbroken.
Remember, continuous means unbroken.
What did differentiable mean?
Differentiable meant that the curve had a tangent line, and
that, in turn, meant that the curve was smooth.
In other words, we may think of differentiable as the
geometric analog of smoothness, continuity as the
geometric analog of unbrokenness.
And now I think it's very easy to see pictorially that a
smooth curve must be unbroken, but an unbroken curve doesn't
have to be smooth.
And in fact, a very trivial illustration of this is the
graph 'y' equals the absolute value of 'x'.
You see, well, what happens here?
At the origin, notice that we have a sharp corner, but that
the curve itself is unbroken there.
You see, I can draw the curve without taking the pencil off
the paper, the chalk off the board.
That's what continuity means from an
intuitive point of view.
Yet I have a sharp corner.
It's not smooth here.
That's why I write smooth in quotation marks.
Hopefully, the textbook, together with our exercises,
will make this much clear to you.
But notice again in terms of the picture how an unbroken
curve doesn't have to be smooth, but a smooth curve has
to be unbroken.
Now, the point is we could say much more
about continuous functions.
I think that all we have to say, though, has
already been said.
It'll be reinforced in the text.
We'll have exercises on this.
But the important thing for now is to understand what we
mean by continuous, how we will use it in the future, and
at any rate, we'll be talking about that more later.
And so until next time, goodbye.
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