The BlancMange Function

Transcription

1 Florida Gulf Coast University The BlancMange Function MAT 4937-Math Senior Seminar Jessica Sobkowiak March 28,2007

2 CONTENTS INTRODUCTION... 3 I. THE BLANCMANGE FUNCTION DEFINED II. CONTINUITY & THE BLANCMANGE FUNCTION III. DIFFERENTIABILITY & THE BLANCMANGE FUNCTION 8-9 CONCLUSION...10 BIBLIOGRAPHY

3 Introduction The purpose of this paper is to construct a function that is continuous everywhere but differentiable nowhere on the real number line. While we know that every function that is differentiable at a point is also continuous at that point, the converse is not always true. Thus, this paper serves as a reminder that continuity need not imply differentiability. We will begin the construction of such a function with a simple piece-wised defined function which will be iterated into sequences that will ultimately produce an infinite sum or infinite series. We will then use the idea of uniform convergence to show that the Blancmange Function is continuous. Finally, we will show that this function, despite its continuity, is nowhere differentiable. I. The Blancmange Function Defined We begin our construction of the Blancmange Function with the piece-wise defined function (figure below) commonly called the Saw-Tooth function: f(x) = { x if 0 x < 1 2 {1-x if 1 2 x < 1 Saw-Tooth Function 3

4 I. The Blancmange Function Defined (Cont d) Next, we will iterate this function forming a sequence of functions: {f n } = { f(2 n-1 x) } 2 n-1 Iterations: First Iteration Second Iteration f 1 = f(2 0 x)/2 0 = f(x) f 2 = f(2 1 x)/2 1 = f (2x)/2 Third Iteration f 3 = f(2 2 x)/2 2 = f(4x)/4 Fourth Iteration f 4 = f(2 3 x)/2 3 = f(8x)/8 Continuing in this fashion, we would iterate n times. Finally, we create the Blancmange Function by summing these iterations from one to infinity. Thus, the Blancmange Function is defined by the infinite series: Ø(x) = f n (x) = f(2 n-1 x) n=1 n=1 2 n-1 4

5 I. The Blancmange Function Defined (Cont d) Thus, the Blancmange Function is constructed by adding together infinitely many iterations of the Saw Tooth Function defined earlier. Upon adding these iterations up, you obtain the following approximate sketch: Originally named the Takagi function after its inventor in 1903, the function was renamed the Blancmange Function after its pudding shape in 1982 (Hairer, 264). II. Continuity & The Blancmange Function In this section, we will establish the continuity of the Blancmange Function by introducing the concept of uniform convergence. By showing that the function converges uniformly, we can use uniform convergence theorem to extend continuity to the function as well. Definition of Uniform Convergence: Let {f n } be a sequence of functions defined on a common domain D. We say {f n } converges uniformly to a function f on D if, for every ε>0, there exists NєN such that f n (x)-f(x) < ε for all n N and xєd. We say f n f uniformly. (Thomson, 393) Notice that N does not depend on the x value chosen. 5

6 II. Continuity & The Blancmange Function (Cont d) When a sequence of functions converges uniformly to a single function, all the functions of the sequence get very close to the function to which they converge. The following picture demonstrates this principle (Thomson, 394): In order to show that the Blancmange Function Is uniformly convergent, we must introduce the Weierstrass M-Test. Theorem (Weierstrass M-Test): If a sequence of positive constants M 1, M 2, M 3,, M n can be found such that in some interval, f n (x) M n for n = 1,2,3,. And M n < (ie, converges to a number), then we can say that f n (x) converges uniformly. (Spiegel, 228) n=1 n=1 Proof: The remainder of the series f n (x) after n terms is: R n (x)= f n+1 (x) + f n+2 (x) +. n=1 Now, R n (x) = f n+1 (x) + f n+2 (x) +. f n+1 (x) + f n+2 (x) +. M n+1 + M n+2 +. But, M n+1 + M n+2 + can be made less than ε by choosing n > N, since M n converges. Since n=1 N is clearly independent of x, we have R n (x) - 0 < ε for n > N and the series is uniformly convergent. QED (Spiegel, 245) Now that we have proved the Weierstrass M-Test, we will use it to show that our function is indeed uniformly convergent. Let M n = {1/(2 n-1 )}. Then M n = (1/(2 n-1 ). Notice n=1 n=1 this is a geometric series with constant ration r = ½ and initial term a = 2. Thus, M n 4 by the geometric series test which says the sum S = a/(1-r). Therefore, since M n <, the second n=1 condition of the Weierstrass M-Test is met. The first condition is also satisfied since f n (x) M n. We can conclude by the Weierstrass M-Test that the Blancmange Function converges uniformly. 6 n=1

7 II. Continuity & The Blancmange Function (Cont d) Now that we have established uniform convergence, we need to use the uniform convergence theorem to extend continuity to the Blancmange Function. Since the Blancmange Function is made up of continuous saw-tooth functions and since it has been shown to be uniformly convergent by the Weierstrass M-Test, the uniform convergence theorem below will guarantee that the Blancmange function itself is continuous everywhere. Theorem (Uniform Convergence): Let{f n } be a sequence of functions defined on an interval [a,b], and let x o є[a,b]. If the sequence {f n }converges uniformly to some function on [a,b] (ie, f n f uniformly), and if each of the functions f n is continuous at x o, then the function f is also continuous at x o. (Thomson, 404) Proof: Let ε<0. We must show that there exists δ>0 such that f(x)-f(x o ) < ε whenever x-x o < δ, xє[a,b]. (Definition of Continuity at a point x o). For each xє[a,b], we have: f(x)-f(x o ) = [f(x)-f n (x)] + [f n (x)-f n (x o )] + [f n (x o )-f(x o )]. Taking absolute value of both sides: f(x)-f(x o ) = [f(x)-f n (x)] + [f n (x)-f n (x o )] + [f n (x o )-f(x o )] f(x)-f n (x) + f n (x)-f n (x o ) + f n (x o )-f(x o ) by Triange Inequality Since f n f uniformly (hypothesis of theorem), there exists NєN such that f n (x)-f(x) < ε/3 for all xє[a,b] and all n N. Thus, f(x)-f(x o ) < f N (x)-f N (x o ) + (2ε)/3. Using the continuity of f N, choose δ>0 such that if xє[a,b] and x-x o < δ, then f N (x)-f N (x o ) < ε/3. Combining above inequalities: f(x)-f(x o ) < ε/3 + (2ε)/3 = ε for each xє[a,b] for which x-x o < δ. Therefore, we have shown that there exists δ>0 such that f(x)-f(x o ) < ε whenever x-x o < δ, xє[a,b]. QED Since the Blancmange Function is made up of continuous functions that converge uniformly to a single function, the Blancmange Function is said to be continuous everywhere by the uniform convergence theorem that we have just proven. 7

8 III. Differentiability & The Blancmange Function In this section we will show that the Blancmange Function is nowhere differentiable by first stating and proving what it means to be differentiable and observing that the Blancmange Function fails to meet this criterion. We know that functions, like the absolute value function, are not differentiable at sharp points; similarly, the Blancmange Function contains sharp points everywhere, causing the function to be nowhere differentiable. Theorem (Differentiability): Let f be differentiable at any aєd f where D f is open. Then for ε>0, there is δ>0 such that for all t 1, t 2, s 1, s 2, t 1 -t 2 < δ, s 1 -s 2 < δ, we have: f(t 2 )-f(t 1 ) - f(s 2 )-f(s 1 ) < ε t 2 -t 1 s 2 -s 1 Proof: By the differentiability of f at a, for each ε>0, there is δ>0 such that: f(x)-f(a) - f (a) < ε/4 x-a if xєb * (δ,a). Thus, there is F(x) defined in B * (δ,a) such that f(x)-f(a) = f (a) + F(x), where x-a F(x) < ε/4. Now, if t 1 a, and thus t 1 єb * (δ,a), we have: f(t 2 )-f(t 1 ) = f(t 2 )-f(a) * t 2 -a f(t 1 )-f(a) * t 1 -a = [f (a) + F(t 2 )] * t 2 -a [f (a) + F(t 1 )] * t 2 -a t 2 -t 1 t 2 -a t 2 -t 1 t 1 -a t 2 -t 1 t 2 -t 1 t 2 -t 1 = f (a) + F(t 2 ) * t 2 -a F(t 1 ) * t 2 -a. Thus, the following is true: t 2 -t 1 t 2 -t 1 f(t 2 )-f(t 1 ) f (a) = F(t 2 ) * t 2 -a - F(t 1 ) * t 1 -a. But, t 1 a t 2 implies t 2 -a < 1 and a- t 1 < 1. t 2 -t 1 t 2 -t 1 t 2 -t 1 t 2 -t 1 t 2 -t 1 From this, it follows that: F(t 2 ) * t 2 -a - F(t 1 ) * t 1 -a = F(t 2 ) * t 2 -a + F(t 1 ) * a-t 1 t 2 -t 1 t 2 -t 1 t 2 -t 1 t 2 -t 1 F(t 2 ) + F(t 1 ) F(t 2 ) + F(t 1 ) < ε/2. Thus, f(t 2 )-f(t 1 ) - f (a) < ε/2. Similarly, if s 1 a s 2, then f(s 2 )-f(s 1 ) - f (a) < ε/2. t 2 -t 1 s 2 -s 1 By Triangle Inequality: f(t 2 )-f(t 1 ) - f(s 2 )-f(s 1 ) < ε. QED. (De Lilo, 794) t 2-t 1 s 2-s 1 8

9 III. Differentiability & The Blancmange Function (Cont d) Now, let a be any real number and f(x)= f n (x). Let δ>0, choose any positive k such that 1 <δ. n=1 2 k-1 Then for each k, there is a uniquely determined integer m satisfying: m a < m+1. 2 k-1 2 k-1 Define: b 1 = m, b 2 = m+1, and b = b 1 +b 2. For example, 1 = 1 < δ. Thus, k=4 and if 2 k-1 2 k a 2, then m=1, b 1 =1, b 2 =2, and b=3 with f 4 (x)= f(8x) The following cases are possible for b i where i=1,2: Case I ( n>k): Since {f n } has period 1, then f n (b 1 )=f n (b)= f n (b 2 )=0. 2 n-1 Thus, f n (b 2 ) - f n (b 1 ) - f n (b 1 ) - f n (b) = 0 b 2 - b 1 b 1 b Case 2 (n<k): Let mєz such that m b 1 < m+1, m b 2 < m+1, and m b<m+1 2 n-1 2 n-1 2 n-1 2 n-1 2 n-1 2 n-1 If m is even: f n (b 2 ) - f n (b 1 ) = f n (b i ) - f n (b) = 1. b 2 - b 1 b i b If m is odd: f n (b 2 ) - f n (b 1 ) = f n (b i ) - f n (b) = -1. b 2 - b 1 b i b It follows then that: f n (b 2 ) - f n (b 1 ) - f n (b i ) - f n (b) = 0. b 2 - b 1 b i b Case 3 (n=k): f k (b 2 ) f k (b 1 ) - f k (b i ) f k (b) = ±1 for i=1,2. b 2 - b 1 b i b Therefore, f(b 2 ) f(b 1 ) - f(b i ) f(b) = f k (b 2 ) f k (b 1 ) - f k (b i ) f k (b) = ±1. b 2 - b 1 b i b b 2 - b 1 b i b Since aє[b 1, b 2 ), then either aє[b 1, b] or aє[b, b 2 ). In either case, the last result shows that the inequality necessary for differentiability at an arbitrary point a: f(t 2 )-f(t 1 ) - f(s 2 )-f(s 1 ) < ε t 2 -t 1 s 2 -s 1 fails if ε 1/2 no matter how small we choose δ. Thus, f is not differentiable at a. Since a was chosen arbitrarily, f is differentiable nowhere. QED (De Lillo, 795) 9

10 Conclusion Finally, we have constructed a function which is continuous everywhere by first forming a sequence of continuous saw-tooth functions and then defining the Blancmange Function as the infinite series of that sequence of functions that we then showed to be uniformly convergent using the Weierstrass M-Test. Since the Blancmange Function is uniformly convergent, we then used the uniform convergence theorem to extend continuity to the Blancmange Function. Next, we defined what it means to be differentiable and showed that for any arbitrary a, that the Blancmange Function failed to meet the criteria to be differentiable at that a. Since a was chosen arbitrarily, we then safely concluded that the Blancmange Function was differentiable nowhere. Thus, the Blancmange Function is both continuous everywhere and differentiable nowhere and serves as a reminder that continuity does not imply differentiability. 10

11 BIBLIOGRAPHY Blancmange Curve. < De Lillo, Nicholas J. Advanced Calculus with Applications. New York: Macmillan Publishing Co., Inc., Hairer, Ernst, and Gerhard Wanner. Analysis by its History. New York: Springer-Verlag, Spiegel, Murray R. Advanced Calculus. New York: McGraw-Hill, Thomson, Brian S., J.B. Bruckner, and A.M. Bruckner. Elementary Real Analysis. UpperSaddle River: Prentice-Hall,

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