Where is f Continuous?

Date: 10/07/97 at 17:03:59
From: Kelly Stone
Subject: High School Calculus

I have a problem in my calculus class. I really need help and I will 
appreciate any help you can give. Here's the problem:

a. Graph the following.

                                           8-|..............
     f(x)= {2^x, if x is rational          7-|          .
           {8  , if x is rational          6-|         .
                                           5-|        .
                                           4-|       .           
                                           3-|     . 
                                           2-|   .
                                           1-|.________________  
                                                 1   2   3   4

b. Besides at x=3, where is is f continuous?

That is the question. I can't get f. Please help!

Date: 10/11/97 at 14:59:56
From: Doctor Chita
Subject: Re: High School Calculus

Hi Kelly:

Have you mistyped your problem? The function as written is supposed to 
be a piece-wise function. However, the domain for both pieces is the 
same - the rational numbers. Usually, a piece-wise function is made up 
of two or more different domains.

I am guessing that the domain of one of the two pieces of your 
function is the set of irrational numbers. For discussion, let's 
assume, then, that

f(x) = {2^x, if x is rational 
       {8, if x is irrational

This means that the domain of f is the set of real numbers, since the 
union of the rational and irrational numbers is the set of real 
numbers.

Consider the two numbers, 1,414, a rational number, and sqrt(2) = 
1.4142... , an irrational number. Both numbers correspond to points on 
the number line, very close together.

According to the new definition of f, then f(1.414) = 2^(1.414) = 
(approximately) 2.665, and f(sqrt(2)) = 8. Consequently, the graph 
of f "jumps" between the exponential curve to the horizontal line at 
these two points.

This scenario will continue as you move along the x-axis, since the 
rational and irrational numbers coexist there. Consequently, f is not 
continuous for any interval in its domain. 

Unlike other types of discontinuous functions, such as g(x) = 1/x that 
has a hole at x = 0, you can't draw a picture of the graph of f 
because there are an infinite number of points jumping between the 
graphs of 2^x and 8. The only way to represent this function is to use 
symbols (the two piece-wise equations) or words.

I hope this answers your question, Kelly. If not, please write back 
and we'll try again.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/