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Many functions, though not all by any means, are defined by a single equation:
(How does this last equation define a function?)
(How does this equation determine a function?)
There are various types of functions, and they can be combined in a variety of ways to produce other functions.It is necessary therefore to spend a fair amount of time at the beginning of this chapter to present these definitions.
If and are two complex-valued functions with the same domain i.e., and and if is a complex number, we define (if is never 0), and by the familiar formulas:
and
If and are real-valued functions, we define functions and by
(the maximum of the numbers and ), and
(the minimum of the two numbers and ).
If is either a real-valued or a complex-valued function on a domain then we say that is bounded if there exists a positive number such that for all
There are two special types of functions of a real or complex variable, the even functions and the odd functions.In fact, every function that is defined on all of or (or, more generally, any function whose domain equals ) can be written uniquely as a sum of an even part and an odd part.This decomposition of a general function into simpler parts is frequently helpful.
A function whose domain equals is called an even function if for all in its domain. It is called an odd function if for all in its domain.
We next give the definition for perhaps the most familiar kinds of functions.
A nonzero polynomial or polynomial function is a complex-valued function of a complex variable, that is defined by a formula of the form
where the 's are complex numbers and The integer is called the degree of the polynomial and is denoted by The numbers are called the coefficients of the polynomial.The domain of a polynomial function is all of i.e., is defined for every complex number
For technical reasons of consistency, the identically 0 function is called the zero polynomial . All of its coefficients are 0 and its degree is defined to be
A rational function is a function that is given by an equation of the form where is a nonzero polynomial and is a (possibly zero) polynomial. The domain of a rational functionis the set of all for which i.e., for which is defined.
Two other kinds of functions that are simple and important are step functions and polygonal functions.
Let be a closed bounded interval of real numbers. By a partition of we mean a finite set of points, where and
The intervals for are called the closed subintervals of the partition and the intervals are called the open subintervals of
We write for the maximum of the numbers (lengths of the subintervals) and call the number the mesh size of the partition
A function is called a step function if there exists a partition of and numbers such that if That is, is a step function if it is a constant function on each of the (open) subintervals determined by a partition Note that the values of a step function at the points of the partition are not restricted in any way.
A function is called a polygonal function , or a piecewise linear function , if there exists a partition of and numbers such that for each is given by the linear equation
where That is, is a polygonal function if it is a linear function on each of the closed subintervals determined by a partition Note that the values of a piecewise linear function at the points of the partition are the same, whether we think of in the interval or (Check the two formulas for )
The graph of a piecewise linear function is the polygonal line joining the points
There is a natural generalization of the notion of a step function that works for any domain e.g., a rectangle in the plane Thus, if is a set, we define a partition of to be a finite collection of subsets of for which
Then, a step function on would be a function that is constant on each subset We will encounter an even more elaborate generalized notion of a step function in Chapter V, but for now we will restrict our attention to step functions defined on intervals
The set of polynomials and the set of step functions are both closed under additionand multiplication, and the set of rational functions is closed under addition,multiplication, and division.
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