Intermediate Value Theorem

#8-oct

Theorem (Intermediate Value theorem).

Let with let be a continuous function. Suppose
or
then there exist some such that

Proof.
Define by

clearly is continuous, also and
using the previous theorem Theorem 4.16 Theorem 4.16

such that
so or

The second case

thus is continues and
using the previous theorem Theorem 4.16 Theorem 4.16

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Corollary.

Let with let be a continuous function then

where is
where is

Proof.

We know that is bounded and attains infimum and supremum
so there exist
Case I:
In this case is constant function and
Case II:

Then consider

case III:
Then consider
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Definition (General interval).

is an interval iff with implies
they are one of these type ...

Theorem.

Let I be a subinterval of let be a continuous function then
is an interval

Theorem (Existance and uniqueness of positive

root).

Fix then for there exist unique such that

Proof.

Then is a polynomial restricted to and hence continuous

Claim 1.

for there exist such that
as so take

Claim 2.

is strictly increases that is if then

Define by tacking
that is

then,

By Intermediate value theorem there exist such that

The uniqueness follows from the strictly increasing property of if and
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#9-oct

Claim.

At any time there are two antipodal points(opposite) in the equator with equal temperature

Proof.
We model the equator by the unit interval 0,1 are identical
we define where is temperature at
we assume that is continuous
we have
Now consider define by

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