U0S2V03 Continuity at a point.txt

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So hopefully you remember that the limit of a function as x
approaches a point, and the value
of the function at the point, don't have to be equal.
But they could be equal.
So we're going to give a name to this situation in which they
happen to be equal.
We're going to say that our function is
continuous at the point x equals a, if the limit of f of x, as x
approaches a equals f of a itself.
Now one great thing about functions
being continuous at points is that if we
know that f is going to be continuous at x
equals a, then that makes calculating this limit really
easy.
We can just calculate f of a instead.
So continuity, when we have it, makes life a lot simpler.
So it's fantastic.
So back to the definition.
We need the limit and the value to be equal.
In particular, they both need to exist.
Graphically, that's going to look like this.
f of a has to exist, so we have to have a point
at x equals a somewhere.
And then the limit as x approaches
a has to equal this same value.
So when we come in from the left,
we have to approach that value, and also coming in
from the right.
Same thing.
It's the overall limit has to equal that value.
So that's basically what being continuous at the point x
equals a has to look like.
There are a variety of ways in which a function might fail
to be continuous at x equals a.
First off, the limit might exist but the value of f at a
might not exist.
That would look like this.
And just keep the same picture just
erase the point at x equals itself.
So we still have a limit, but now f of a doesn't exist.
This is no longer continuous.
And we have a hole at x equals a.
Another way to be discontinuous would
be if the limit exists and then f of a also exists,
but they're not equal.
That would look just like this, where
we put a value for f of a that's different than the limit
as x approaches a.
So we still have the hole in the graph of f,
and that's a discontinuity.
Finally, there are the cases where f of a exists,
but the limit does not exist.
So we know that there are a lot of ways
that a limit might not exist.
For instance, it might blow up as we come in from one side.

One case I want to point out specifically
is where the limit from the left exists,
and the limit from the right exists,
but they're just not equal.
So when they don't match, the overall limit doesn't exist,
and the function is not continuous,
we get this jump at x equals a.
Something to notice here-- we don't have an overall limit,
but we do have the limit of f of x as x
approaches a from the right.
And that does happen to be equal to f of a.
So when that happens, then we say that f
is right-continuous at a.
And we can have a similar definition for left-continuous.
I'm not going to write that down.
So in this picture f is right-continuous,
but not left-continuous at x equals a.
But to be continuous overall, we need
to be both right- and left-continuous
at the point in question.
We're not here so instead we have this jump,
we have a discontinuity.
So to summarize, if you're discontinuous at a point,
then either the graph is going to jump there,
or maybe it has a hole-- we saw that earlier-- or coming in
from one side, the function blows up or goes crazy somehow.
But If your function is continuous at x equals a, that
means none of that stuff happens,
and instead we have just a nice happy function
right at this point.
And not only is the picture happy,
but it makes computing limits easy.
And so most of the rest of the sequence
is going to be about how can we recognize just from a formula,
whether a function is going to be continuous.
Sound good?