U0S2V08 Limit Laws and Continuity.txt

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In this video, we'll see what the limit laws tell us
about the continuity of functions.

Let's say we have two functions, f and g, which
are continuous at a point a.
By definition, that means that there are limits,
as x approaches a, match their values at a.
And suppose that we have h of x equaling f of x times g of x.
What can we say about the continuity of h?

Well, the limit of a product is the product of the two limits.
And so as x approaches a, f of x is approaching f of a,
and g of x is approaching g of a.
So h of x is approaching f of a times g of a, which is h of a.
So the limit of h of x as x approaches
a equals the value of h at a.
In other words, h is continuous at a.

So what we've just proven is that if f and g are
continuous at some point, then their product
is continuous at that point.
And if f and g are continuous everywhere,
then their product will be continuous everywhere.
And the same thing will go for their sum or their difference.
We just use the other limit laws in exactly the same way.

We can deduce that a lot of functions
are continuous, based on these principles.
For instance, 5 is a constant function,
so that's continuous everywhere.
And x we know is a function that's continuous everywhere,
so 5 times x will be continuous everywhere.
And then we can take 5x times x, and that's
the product of 2 continuous function, so it'll
be continuous, and that equals 5x squared.
And anything like that.
We can even add these things, so 5x squared plus x.
That's the sum of continuous function, so it's continuous.
Basically any polynomial will be continuous
on the entire real line.

What else can we do?
Well, we had a limit law for division, part one anyway.
In the case where the limit of the denominator was not 0.
And that tells us that if f and g are continuous,
then the quotient f over g will be continuous also,
as long as we're talking about points where g is not zero.
In other words, as long as the quotient is defined.
So for instance, tangent is sine x over cosine x.
And that's the quotient of two functions, which are continuous
everywhere.
So the tangent function will be continuous at all points
where cosine x is not 0.
And if you remember the graph of the tangent function,
this should make sense.
The graph looks like this, and it's continuous
everywhere except for these points where there are breaks,
and that's minus pi over 2, pi over 2, et cetera.
Exactly the places where cosine is zero.
If on the other hand, we take the function
1 over x squared plus 1, here we have a polynomial
over a polynomial, so again, the quotient of two functions,
which are continuous everywhere, but this denominator
is never zero.
So this quotient is going to be continuous everywhere.
No exceptions.

There's one other thing that we should discuss,
and that's the composition of functions.
If f and g are continuous everywhere,
what can we say about h of x, which
is equal to f of g of x, the composition of f with g?
Well, as x approaches some value a,
we know g of x is going to be approaching g of a,
and that's just because we know g is continuous.
But we also know that f is continuous at g
of a, so we've got some value approaching some point.
That means f of this value is going to be
approaching f of that point.
f of g of x is going to be approaching f of g of a.
And we can translate that as h of x approaching h of a.
In other words, h is continuous at this point a.
So up above, we can say that if f and g are continuous
everywhere, then f composed with g
will also be continuous everywhere.
And this little circle if you've never seen it before,
that just denotes composition.
So for instance, if we take sine of x squared plus 1,
here we have the composition of two functions,
the inner function is x squared plus 1, that's a polynomial,
so it's continuous everywhere.
And the sine function, the outer function, we
know that's continuous everywhere,
so this overall function is continuous everywhere.

Alright.
Well we've discussed addition, subtraction, multiplication,
division, and now composition.
And those are all of the common ways
that we'll be using to build up complicated functions.
So we should be good to go.