— 6.10. Global properties of continuous functions —
Theorem 51 (Intermediate Value Theorem) Let and is a continuous function. Let such that , then there exists such that .Proof: Omitted. |
Intuitively speaking the previous theorem states that the graph of a continuous function doesn’t have holes in it if the domain of the function doesn’t have any holes in it too.
Corollary 53 Let , a continuous function. Then is also an interval.Proof: Let and . By definition of infimum and supremum it is . Using Theorem 51 it is .Thus we have the following four possibilities for : |
As an application let us look into with odd and . It is for large (positively or negatively) values of . It is and .
Now
- is a continuous function.
- The domain, of is which is an interval.
- and , implying
By corollary 52 it is . Which means that every odd polynomial function has at least one .
Theorem 54 (Continuity of the inverse function) Let be an interval in and a continuous function and strictly monotonous. Then is continuous and strictly monotonous.Proof: Omitted. |
This theorem has important applications since it allows us to define the inverse functions of the trigonometric functions.
— 6.10.1. Arcsine function —
In the function is injective:
Hence one can define the inverse of the sine function in this suitably restricted domain.
Where denotes the inverse of .
Since it is . Also by theorem 54 is continuous.
The graphical representation of is
and it is evident by its representation that is an odd function.
— 6.10.2. Arctangent function —
In the function is injective:
Hence one can define the inverse of the tangent function in this suitably restricted domain.
Where denotes the inverse of .
Since it is . Also by theorem 54 is continuous.
The graphical representation of is
and it is evident by its representation that is an odd function.
— 6.10.3. Arccosine function —
In the function is injective:
Hence one can define the inverse of the cosine function in this suitably restricted domain.
Where denotes the inverse of .
Since it is . Also by theorem 54 is continuous.
The graphical representation of is
Another way to define the arccosine function is to first use the relationship
to write
— 6.10.4. Continuous functions and intervals —
Theorem 55 (Extreme value theorem) Let and . Then has a maximum and a minimum.Proof: Let be the codomain of and .By Theorem 17 in post Real Analysis – Sequences II there exists a sequence of points in such that .
Since the terms of are points of for each there exists such that . Since is a sequence of points in the compact interval (see definition 22 in post Real Analysis – Sequences IV) , by Corollary 27 (also in post Real Analysis – Sequences IV) there exists a subsequence of that converges to a point in . Let such that . Since is continuous in it is, by definition of continuity, (see definition 34) . But , which is a subsequence of . Since it also is . But . In conclusion it is , hence . That is . For the minimum one can construct a similar proof. This proof is left as an exercise for the reader. |
One easy way to remember the previous theorem is:
Continuous functions have a maximum and a minimum in compact intervals.
Theorem 56 Let be a compact interval of and continuous. Then is a compact interval.Proof: By corollary 53 is an interval.By theorem 55 has a maximum and a minimum.
Hence is of the form . Thus is a limited and closed interval, which is the definition of a compact interval. |
One easy way to remember the previous corollary is:
Compactness is preserved under a continuous map.