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/ |
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