U0S2V06 Overall continuity.txt

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In our last video, we talked about what
it meant for a function to be continuous at a point.
Now, we know that a function can be
continuous at some points but not others.
For instance, if we take this function,
then it's continuous at this point
and it's continuous at this point,
but it's not continuous here.
There's a jump.
And it's not continuous here, either.
There's a hole.
Now if a function happens to be continuous not just at a point,
but at every point, then we'll say
that the function is continuous on the real line,
or continuous everywhere.
And most of the time, if we just say
continuous without specifying a point, this is what we'll mean.

This means that there are no discontinuities anywhere.
If we draw the graph from left to right,
we never have to lift our pen for a jump or a hole
or anything like that.
It's just nice and connected.

Continuous functions are great, especially from the point
of view of limits, because the limit of a continuous function
will always equal the value of the function at that point.
Let's look at a couple of basic functions.
First the constant function f of x equals 3.
So here it is.
And we asked you to think about this one.
And hopefully you realize that this function, and indeed
any constant function, will be continuous at any point.
So that's our first example of a function
that's continuous everywhere, the constant functions.
Let me give you a few other basic functions that
are continuous everywhere.
We're not going to prove that they are,
but you should feel free to use the fact
that these are continuous.
We can take g of x equals x.
And this, too, is continuous overall.
Here's its graph.

How about the absolute value function?
That's continuous on the entire real line.
Its graph looks like this.
No holes, no jumps.
It's perfectly continuous.
Another function that's continuous everywhere
is sine x.
And its graph looks like this.
Nice and lovely.
No discontinuities there either.
There's also cosine x.
I'm not going to draw that one.
However, the tangent function is not continuous everywhere.
Remember that tangent x is sine x over cosine x.
And so the tangent function is not
defined at places where cosine x is 0.
And if it's not defined at those points,
then it certainly cannot be continuous there.
Now, it does turn out that the tangent function is continuous
everywhere else.
But it's not continuous everywhere.
So many of our basic functions are continuous everywhere,
but you do have to be careful because not all of them are.
We have a few quick questions for you,
and then we'll use the limit laws
to see how combining these basic functions affects continuity.
So stick around.