In this video, I will present the definition of continuity.
Actually, I will do a bit more:
I will write TWO equivalent, rigorous definitions of continuity;
I will present some non-rigorous ideas
that explain the intuition behind the concept; and I will explain
how we define continuity differently in different domains.
I will begin with a handwavy idea.
When we say that a function "is continuous”,
we mean that we can "sketch its graph in one go”;
without lifting the pen from the paper.
This is, of course, not rigorous, but it helps.
For example, this graph CAN be sketched without lifting the pen from the paper.
I want to transform this vague idea into a rigorous definition.
Let’s examine some examples of graphs that CANNOT be drawn “in one go”,
and see why.
This graph, for example,
CANNOT be sketched without lifting pen from paper,
because it has a hole.
Notice that it does not matter
whether the function is UNdefined at the point,
or it is defined “with the wrong value," so to speak.
Either way, I want to say that this function is not continuous.
This example is different.
It has a jump, so I cannot sketch it without lifting pen from paper, either.
Neither can I this one, because it has a vertical asymptote.
And finally, I cannot sketch this one without lifting pen from paper, either.
I can sketch the graph fine on EITHER side of x = 0,
but I CANNOT get through 0, because the function oscillates wildly.
I want to say that all these examples are not continuous.
Many of these examples of bad functions have one thing in common:
they lack a limit.
This function does not have a limit as x approaches 0.
THIS function does not have a limit as x approaches 0.
and THIS function does not have a limit as x approaches 1.
However, THIS example is different.
The function in this graph HAS a limit as x approaches a.
But it has a hole, because the function is undefined.
On the other hand, THIS function has a limit as x approaches a,
AND it IS defined at a.
However, the limit and the value are different,
and thus it still has a hole.
Let's put all these examples together.
This suggests how I want to define continuity.
I will say that a function f is continuous at a
when the limit of f(x) as x approaches a
is equal to the value of the function f at a.
All the examples I constructed,
where I could not sketch the graph of the function
without lifting pen from paper, failed THIS,
so this seems like a good definition.
To make this into a precise definition, I have to introduce my variables,
I will say that a is a real number,
and I will require f to be a function
defined at least on an interval centered at a,
so that I can study its limit at a.
To be very explicit, this definition means three things.
1. The limit must exist and therefore be a number.
2. The function must be defined.
And 3. The limit must be equal to the value of the function.
Notice that both the limit and the function must be the same NUMBER.
So for example,
if they both do not exist, then they do not satisfy the definition.
And there we have it! This is the definition of continuity.
This was the simplest way to write the definition,
but there is an alternative, equivalent way:
we can use the (ε, δ)-definition of limit EXPLICITLY.
Here is what happens if we do it.
First, as a reminder, this is the definition
of “the limit as x approaches a of f(x) is L”.
For a function to be continuous, the limit must not be just any number L,
but specifically f(a).
So let’s simply change L into f(a).
And now we can do a simplification.
In the definition of LIMIT we must exclude the case x = a,
because we don't even CARE whether the function is defined at a.
but when we want CONTINUITY, the function MUST be defined at a,
and the limit must be f(a), so we NO LONGER need to exclude x = a.
This condition at the bottom, when we remove the exclusion x = a,
is entirely equivalent.
And there we have it!
This is the SECOND definition of continuity,
which is ENTIRELY EQUIVALENT to the first one.
It may be called the (ε, δ)-definition of continuity.
It may APPEAR more complicated,
but only because the first definition hides the εs and δs inside the limit.
In any case, it is useful to have multiple equivalent definitions.
In different contexts, either one or the other may come handy.
This process has highlighted an idea that, while NOT rigorous,
is extremely useful to keep in mind.
Here is, in few words, the difference between merely asking for a limit,
and asking for continuity.
When we say that lim_(x → a) f(x) = L, roughly, we mean
that IF x is close to a, BUT NOT a, THEN f(x) is close to L.
But when we say that f is continuous at a,
roughly, we mean that IF x is close to a, INCLUDING a,
then f(x) is close to f(a).
THAT is the difference.
So far, I have explained what it means
for a function to be continuous at a point. Here it is again.
But most of the time we want functions to be continuous on a whole domain.
In the final part of the video, I will explain what that means.
If we say that a function is continuous on an OPEN interval,
we simply mean that it is continuous AT EVERY POINT on the interval.
However, if we say that a function is continuous on a CLOSED interval,
it is a bit different.
Look at this example.
I want to say that
the function in this graph is continuous on the closed interval [a,b],
because I certainly can sketch the graph without lifting pen from paper.
But I CANNOT say that the limit exists at the endpoint a,
according to the definition we are using,
because the function is NOT even DEFINED to the left of a.
So instead, I will require the following:
at all the points in the interior of the interval,
I require the function to be continuous.
at the left-endpoint a,
I only need the RIGHT-SIDE limit of f(x) to be equal to f(a).
And similarly, at the right-endpoint b,
I ONLY need the LEFT-SIDE limit of f(x) to be equal to f(b).
And that's it.
That is how we define being continuous on an open interval,
and being continuous on a closed interval.
Of course, there are other types of domains,
but you can figure out the corresponding definitions.
To be fair, there is a better way to define "continuous on a domain"
that works for ALL domains at once.
And then, we don't HAVE to break it into cases.
In further math courses,
for example if you study multivariable calculus,
you will need that better definition.
But for our purposes this simple and this is enough.
And finally, one more concept:
if we ever simply say that a function is continuous
without specifying WHERE,
we mean “continuous on its domain”.
Careful, though!
Because things like the following may happen.
Look at the function in this graph.
It IS continuous,
if by "continuous," we mean "continuous on its domain."
But it is NOT continuous at 1.
This may SOUND like a contradiction,
but it IS not, if you pay attention to the precise definitions
that we have given to the concepts.