U0S2V11 IVT.txt

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Here's our set up.
We have two points on the graph of some function f.
And we have a horizontal line y equals
M, such that the two points are on opposite sides of the line.
So I've drawn it here where the first point is below the line
and the second point is above the line,
but it could be the other way around, doesn't matter.
So the question that we want to answer in this video is,
as the graph of f travels from here to here,
does it have to intersect this line?
The answer is yes if we know that the function is
continuous.
So if f is continuous then it has no holes and no jumps.
So as we go from this point to this point,
there's got to be at least one place where the graph of f
crosses this line.
Now it could be that there's more than one point.
The graph of f could do something like this,
but we know for certain that there's
got to be at least one point.
This is called the Intermediate Value Theorem.
And it states that if f is continuous
and we have some value M that's between the values of f
of a and f of b, in other words, M
is an intermediate value then, well, then what?
Well, what we're saying is that there's
at least one point, which I'm going to call c in the middle
here, where this graph hits the line y equals m.
So in other words, f of c equals M. So that's
what we're going to write above.
That there is at least one point c between a and b such
that f of c equals M, but we need these conditions.
In order to use the Intermediate Value Theorem
M should be in between f of a and f of b and very important
f needs to be continuous.

Well, I should say that we don't actually need
f to be continuous everywhere.
It doesn't matter if f has a discontinuity way out
here to the left, for instance.
We only need f to be continuous on this stretch from a to b.
So we'll say if f is continuous on the open interval a comma b,
and then, we also need it to connect to this point.
So it should be right continuous at a
and then similarly, we need it to connect to this point.
So we need that it's left continuous at b.

So that's the complete Intermediate Value Theorem.
If f is continuous in the right ways at the relevant places
and M lies in between these two values of f,
then f is going to equal M at at least one place.