Real-Valued Functions of One Real Variable

Abstract

Consider a function $f: A \rightarrow B$. As we stated earlier, by this we mean that for every element a of the set A, there exists a corresponding b ∈ B, which is denoted by b = f(a).

Appendix: Basics of Coordinate Geometry

Let us consider two perpendicular lines in the plane, the first of which we call the x-axis, the second the y-axis. We call the intersection of the two axes the origin. We imagine each axis to be a number line, that is, for every point on each axis, there is a corresponding real number that gives the distance of the point from the origin; positive in one direction and negative in the other.

If we have a point P in the plane, we get its projection onto the x-axis by taking a line that contains P and is parallel to the y-axis, and taking the point on this line that crosses the x-axis. The value of this point as seen on the number line is called the first coordinate of P. We get the projection onto the y-axis similarly, as well as the second coordinate of P. If the first and second coordinates of P are a and b respectively, then we denote this by P = (a, b). In this sense, we assign an ordered pair of real numbers to every point in the plane. It follows from the geometric properties of the plane that this mapping is bijective. Thus from now on, we identify points in the plane with the ordered pair of their coordinates, and the plane itself with the set $\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}$. Instead of saying “the point in the plane whose coordinates are a and b respectively,” we say “the point (a, b).”

We can also call points in the plane vectors. The length of the vector u = (a, b) is the number $\vert u\vert = \sqrt{a^{2 } + b^{2}}$. Among vectors, the operations of addition, subtraction, and multiplication by a real number (scalar) are defined: the sum of the vectors (a ₁, a ₂) and $(b_{1},b_{2})$ is the vector $(a_{1} + b_{1},a_{2} + b_{2})$, their difference is $(a_{1} - b_{1},a_{2} - b_{2})$, and the multiple of the vector (a ₁, a ₂) by a real number t is $(ta_{1},ta_{2})$.

Addition by a given vector $c \in \mathbb{R}^{2}$ appears as a translation of the coordinate plane: translating a vector u by the vector c takes us to the vector u + c. If $A \subset \mathbb{R}^{2}$ is a set of vectors, then the set $\{u + c: u \in A\}$ is the set A translated by the vector c.

If c = (c ₁, c ₂) is a fixed vector that is not equal to 0, then the vectors $t \cdot c = (tc_{1},tc_{2})$ (where t is an arbitrary real number) cover the points on the line connecting the origin and c. If we translate this line by a vector a, then we get the set $\{a + tc: t \in \mathbb{R}\}$; this is a line going through a.

Let a and b be distinct points. By our previous observations, we know that the set $E =\{ a + t(b - a): t \in \mathbb{R}\}$ is a line that crosses the point a as well as b, since $a + 1 \cdot (b - a) = b$. That is, this is exactly the line that passes through a and b. Let a = (a ₁, a ₂) and $b = (b_{1},b_{2})$, where $a_{1}\neq b_{1}$. A point (x, y) is an element of E if and only if

$$\displaystyle{ x = a_{1} + t(b_{1} - a_{1})\ \ \mbox{ and}\ \ y = a_{2} + t(b_{2} - a_{2}) }$$

(9.10)

for a suitable real number t. If we isolate t in the first expression and substitute it into the second, we get that

$$\displaystyle{ y = a_{2} + \frac{b_{2} - a_{2}} {b_{1} - a_{1}} \cdot (x - a_{1}). }$$

(9.11)

Conversely, if (9.11) holds, then (9.10) also holds with $t = (x - a_{1})/(b_{1} - a_{1})$. This means that (x, y) ∈ E if and only if (9.11) holds. In short we express this by saying that (9.11) is the equation of the line E.

If a ₂ = b ₂, then (9.11) takes the form y = a ₂. This coincides with the simple observation that the point (x, y) is on the horizontal line crossing $a = (a_{1},a_{2})$ if and only if y = a ₂. If $a_{1} = b_{1}$, then the equation of the line crossing points a and b is clearly x = a ₁.

For $a,b \in \mathbb{R}^{2}$ given, every point of the segment [a, b] can be obtained by measuring a vector of length at most | b − a | from the point a in the direction of b − a. In other words, $[a,b] =\{ a + t(b - a): t \in [0,1]\}$.

That is, $(x,y) \in [a,b]$ if and only if there exists a number $t \in [0,1]$ for which (9.10) holds. In the case that $a_{1} < a_{2}$, the exact condition is that $a_{1} \leq x \leq a_{2}$ and that (9.11) holds.