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Hierarchy of Functions
Jillian Gaglione
February 4, 2004
Contents
1 Introduction 1
2 Lp^ Functions 1
3 Bounded Functions 2
4 Continuous Functions 2
5 Uniformly Continuous Functions 3
6 Compact Sets 3
7 First Mean Value Theorem for Integrals 4
1 Introduction
Functions satisfy different properties which allow us to learn more about them. More specifically, we will be focusing on varying types of functions that lie in a single real or complex plane. These functions are know as Lp^ functions, bounded functions, continuous functions, and furthermore, uniformly continu- ous functions. We will be defining what we mean by these terms as well as showing some examples to understand these properties.
2 Lp^ Functions
Definition: Let p > 0 and |f (x)|p^ be integrable over [a, b]. Then this function is known as Lp[a, b]. In simplier terms, an Lp^ function can be defined as a function where (^) โซ b
a
|f (x)|p^ dx
exists.
Example 2.1 Let f (x) = (^1) x on the interval [0.1]. Show if (a) f โ L^1 , (b)
f โ L^2 , and (c) f โ L
(^12) on the interval [0, 1]:
(a)
0
x
โฃ (^) dx = ln(x) over [0, 1]. We know that the ln(x) from [0, 1] is
undefined because the ln(0) = โ.
(b)
0
x^2
โฃ (^) dx = โ 1 x over^ [0,^ 1]. We know that^ โ^
1 x from^ [0,^ 1]^ is undefined because โ 10 is undefined.
(c)
0
x 12
โฃ dx = โ 12 x
(^12) over [0, 1]. We know that this integral evaluated at
[0, 1] is equal to โ 12.
Therefore, f 6 โ L^1 and f 6 โ L^2 , but f โ L (^12)
3 Bounded Functions
Definition: Suppose S is a nonempty set. Then the function S โ M is a bounded function if there exists a M such that |f (x)| < S for all x โ S.
The set of all functions which are bounded on S is denoted by B(S) and B(S) is a linear space.
Example 3.1 The set y = sin(x^2 ) is bounded on โโ < x < โ.
Example 3.2 The Gamma Function y = ฮ(x) is not bounded on the interval 0 < x โค 1 and the interval 1 โค x < โ.
4 Continuous Functions
Definition: A function is continuous at a point c in its domain D if given any > 0 there exists a ฮด > 0 such that if x โ D and |x โ c| < ฮด, then |f (x) โ f (C)| < .
Moreover, a function is continuous if it is continuous at every point in its domain D.
Example 4.1 The function f (x) = 5xโ 6 is continuous in its domain However, the function
g(x) =
1 , if x = rational 0 , if x = irrational
is not a continuous function.
7 First Mean Value Theorem for Integrals
Definition: Let f (x) be continuous on [a, b], and let
F (x) =
โซ (^) x
a
f (t)dt
Then, we know from the Fundamental Theorem of Calculus that F โฒ(x) = f (x). We also know from the Mean Value Theorem that there exists a c โ (a, b) such that F (b) โ F (a) b โ a
= F โฒ(c) โ F (b) โ F (a) = F โฒ(c)(b โ a)
This implies that โซ (^) b
a
f (t)dt = f (c)(b โ a)
and is known as the First Mean Value Theorem for Integrals. And the number f (c) is referred to as the weighted average value of f on the interval [a, b].
Example 7.1 Find the average value of f (x) = 5 โ 2 x on the interval [โ 1 , 2] and find a point in the interval at which the function takes on this average value.
Solution (^) bโ^1 a
โซ (^) b a f^ (x)dx^ =^
1 2 โ(โ1)
12 (5^ โ^2 x)dx^ =^ 1 3 [5x^ โ^ x
2 ]โ 12 = 1
3 (6^ โ
f (x) = 4 โ 5 โ 2 x = 4 โ โ 2 x = โ 1 โ x =
Therefore the average value of f (x) on [โ 1 , 2] is 4 and a point in the interval at which the function takes on this average value is 12.
References
.D. Aliprantis, O. Burkinshaw. Principles of Real Anal- ysis, 2nd ed., Academic Press, 1990. Khamsi, Mohamed A. S.O.S Mathematics, Math Medics LLC, 1999-2003. Availablehttp://www.sosmath.com/calculus/integ/integ04/integ04.html Wachsmuth, Bert. Interactive Real Analysis, 1994-2000. Availablehttp://www.shu.edu/projects/reals/
