In this video we're going to look at the key difference between continuity and uniform continuity.
So we say function is continuous on the set of real numbers.
If for every no C in the set of real numbers and for every Epsilon greater than zero there exists some
Delta greater than zero.
Notice delta here can depend
on C and epsilon so it could depend on both the value of c and the value of epsilon right.
They are stated before the delta is stated such that for every real number x whenever the distance between
X and C is less than Delta the distance between f of x and f of C is less than epsilon.
So in the definition of continuity your delta is allowed to depend on the value of x.
In this case we called it C and on Epsilon now F is uniformly continuous on the set of real numbers
f for every Epsilon greater than zero.
We can have some Delta greater than zero now here no mention of c or x is made prior to the delta.
So here a Delta can only depend on Epsilon.
It's not allowed to depend on any value of x or C no value of extra C is mentioned in the definition
prior to Delta.
Such that for every x y in the set of real numbers whenever the distance between X and Y is less than
Delta we have that the distance between f of x and f of Y is less than epsilon.
So the key difference that you'll see if you if you look this up on the Internet or you look in a book
is that in continuity the value of Delta can only depend on C and epsilon so it could depend on both
and in uniform continuity.
Delta only depends on epsilon.
So a uniform continuity only on Epsilon and in continuity.
It's allowed to depend on both.
Let's break this down a little bit further.
So we're gonna take this condition here in the red box notice that implies this condition here.
Notice this condition here and uniform continuity is the same right except here.
We used a y value instead of a C. likewise this is very similar in fact almost identical to the continuity
one except again.
Here we used Y instead of C..
So let's go ahead and write down the conditions for continuity and break it down further.
So we have the absolute value of X minus C less than Delta.
And that should imply that the absolute value of F of X minus F and see is less than epsilon so you
can drop the absolute value and when you do that you get X minus C less than Delta greater than negative
delta.
You can do that because you can think of absolute value as distance so the absolute value of X minus
C is the distance between X minus C and zero.
So this is a number and it's distance between distance B from zero is less than Delta.
That means it must be between negative delta and Delta.
So again you have a number who's distance from zero is less than Delta so it should be between negative
delta and Delta.
We can solve this for X by adding C to all three sides when we do that it can be written as C minus
delta less than X less than C plus Delta.
And this implies and again here you can drop the absolute value so you get F of X minus f of C less
than epsilon and greater than negative Epsilon you can add FC to all three sides and we get F of X less
than f of C plus epsilon greater than and then see if I can squeeze it in F of C minus epsilon so this
condition here implies this condition here.
Okay now let's carefully graph this an attempt to explain this graphically in my pictures aren't the
best.
So when you go really slow.
So there is the y axis and then make it bigger and all the room I can get.
And then here's the x axis.
This is X and this is why.
Okay and then maybe C is down here.
This is our C and this year could be C plus Delta.
And this he would be C minus delta then maybe this is FFC and this here would be f of C plus epsilon
and then here we have f of C minus epsilon okay.
Let's talk about what's actually going on.
So this condition let me use a different color.
Let's try.
GREENE So this condition here is saying that x is between C minus delta and C plus Delta.
So whenever X is in the green.
So here this shaded green so if you're x value is in the green then your y value.
So F of X that's your y value right.
Has to be between f of C minus epsilon.
So this number here and F of C plus epsilon.
So again whenever your x is in the green down here your Y has to be in the green up here.
Right.
It has to be between these y values.
So what you could do is you could draw a rectangle right you could draw a rectangle you could visualize
it as a rectangle if you like.
And again it's saying that whenever X is in the green your y values also have to be in the green.
So when I draw my function here here's FFC I have to stay within the green for the y values.
So I can't I can't go up here.
I can't do that.
That's bad right.
I have to stay down in the green.
I'm still in the green still on the green still in the green.
I'm out.
OK.
So when I say in the green I mean here in the blue.
So again whenever your ex is down here in the green.
Think of your why is the blue your why values have to be between these.
So notice my y values are okay.
So what's the difference between continuity in and uniform continuity in terms of this rectangle.
So in uniform continuity we know that the Delta can only depend on Epsilon right.
So this rectangle is the same for the entire function.
So in uniform
continuity
this rectangle
works
for the entire function so at every x value you can draw this rectangle.
Right.
This this the same exact rectangle.
And whenever X is in the green Y is in the blue right.
No matter what.
Right.
So at any point on the graph you can put this rectangle there and center it at that point and it will
work.
It will satisfy the definition of uniform continuity.
Hence the name uniform.
The continuity is uniform throughout because the rectangle is always the same.
It's always the same rectangle.
Right.
You might wonder well when could this fail.
Right.
When when when would you not have uniform continuity.
Well if you have something like this everything looks good then all of a sudden you have a function
that does this.
Well you see the rectangle might change for example.
Let me draw a different picture here.
Say we have a function here.
And it's tough to draw rectangles all the same size so let's say we have a rectangle here so everything
is OK here.
Right.
So where we're within the blue right where within the blue when the y values but then let's say my function
does this.
If I draw the same rectangle down here
you see the red line escapes right escapes the rectangle.
Right.
So let's go back to our green and blue analogy right.
So whenever you're in the green the y values have to be in the blue but that's not true right because
look right here means blue here the y values escape down here you see these y values here are not where
they're supposed to be right they escape the rectangle.
So the function has to come in through one side of the rectangle and come out through the other side.
It can't it can't escape like this.
This is bad right.
This is bad.
So if you tried to use the same rectangle on continuity it wouldn't work.
So in continuity what we have is we have different rectangles.
Right.
And we can have different rectangles because Delta right depends on the value of c or x and epsilon.
So the size of the rectangle will change in continuity whereas in uniform continuity it's the same rectangle
throughout.
So again the key idea here is graphically in uniform continuity.
It's the same rectangle hence the name uniform right.
The continuity is uniform and continuity
the rectangle can change the rectangle can change.
So the continuity is not uniform I wrote that really really fast.
It's just it's exciting to talk about uniform continuity.
I hope this video has made sense.
That's it.
Thanks for watching.