# MATH 409 Lecture 11

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## Continuous Functions

### Review

**Theorem.** Any polynomial of odd degree has at least one real root.

*Proof.* Let be a polynomial of positive degree . Note that . For any , we have

which converges to 1 as . As a consequence, there exists such that if . In particular, the numbers and are of the same sign if . In the case is odd, this implies that one of the numbers and is positive while the other is negative. By the Intermediate Value Theorem, we have for some .

### Continuity Over Domain

**Theorem.** Given a function and a point , let denote the restriction of to the interval and denote the restriction of to .

The function is continuous if and only if both restrictions and are continuous.

*Proof.* For any , the continuity of at is equivalent to the continuity of at . Likewise, the continuity of at a point is equivalent to the continuity of at . The function is continuous at if as . The restriction is continuous at if as . The restriction is continuous at if as . Therefore is continuous at if and only if both and are continuous at .

For example, the function is continuous on . Indeed coincides with on and with the function on .

### Continuity of Compositions

**Theorem.** Let and be two functions. if , then is a well-defined function on .

If is continuous at a point and is continuous at , then is continuous at .

*Proof.* Let us use the sequential characterization of continuity. Consider an arbitrary sequence converging to . We have to show that

Since the function is continuoous at , we obtain that as . Moreover, all elements of the sequence belong to the set . Since the function is continuous at , we obtain that as .

#### Examples

If a function is continuous at a point , then a function , , is also continuous at .

Indeed, the function is the composition of with the continuous function .

If functions are continuous at a point , then functions and are also continuous at .

- Indeed, for all , and
- for all

##### Trigonometric Functions

For a unit circle with angle measured ccw from the point , we define

**Theorem.** for all .

These are usually proved using areas, but we will use length.

*Proof.* The length of is the length of the line from to the -axis, is the length of the arc from to , and is the length of the segment from to , where is parallel to .

We approximate the length of an arc by summing the lengths of a bunch of lines next to the arc (similar to how was first estimated)

**Theorem. [Continuity of sine].** is continuous for all .

We know that for all . Since , we get if . In the case , this estimate holds too as . Now using the trigonometric formula,

We obtain

It follows that as for every . That is, the function is continuous.

**Theorem. [Continuity of cosine.]** is continuous for all

*Proof.* Since for all , the function is a composition of two continuous functions and .

**Theorem. [Continuity of tangent].** is continuous for all

*Proof.* Since , the function is continuous on its tentire domain

and for .

*Proof.* Since and the identity functions are continuous, it follows that if is continuous on . Further, we know that for . Therefore Since , the squeeze theorem implies that as . The left-hand limit at is the same as for all . Thus the function is continuous at as well.

### Monotone Functions

Let be a function defined on a set .

The function is **increasing** if, for any , implies .

It is called **strictly increasing** if implies .

Similarly, is **decreasing** if, for any , implies .

It is called **strictly decreasing** if implies .

Increasing and decreasing functions are called **monotone**, and strictly increasing and strictly decreasing functions are called **strictly monotone**.

**Theorem.** Any monotone function defined on an open interval can have only jump discontinuities.

**Theorem.** A monotone function defined on an interval is continuous if and only if the image is also an interval.

**Theorem.** A continuous function defined on a closed interval is one-to-one if and only if it is strictly monotone.

### Continuity of Inverse Functions

Suppose is a strictly monotone function defined on a set . Then is one-to-one on . Then is one-to-one on so that the inverse function is a well-defined function on .

**Theorem.** If the domain of a strictly monotone functino is a closed interval and is continuous on , then the image is also a closed interval, and teh inverse function is strictly monotone and continuous on .

*Proof.* Since is continuous on the closed interval , it follows from the extreme value and intermediate value theorems that is also a closed interval. The inverse function is strictly monotone since is strictly monotone. By construction, maps the interval onto the interval , which implies that is continuous.

### Examples

**Power function.** for and .

The function is continuous on . It is strictly increasing on the interval anf . In the case is odd, the function is strictly increasing on and . We conclude that the inverse function is a continuous function on if is even and a continuous function on if is odd.

for and .

The function is strictly decreasing on . It is continuous on and maps this interval onto itself. Therefore the inverse function is a continuous function on .