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MY FAVORITE FUNCTIONS

MY FAVORITE FUNCTIONS. OR Continuous from what to WHAT?!?. Section 1. Accordions. y = sin x . _ x. 1. y = sin. f(x) = x sin. . 1. x. f is continuous at 0 even though there are nearly vertical slopes. g(x) = x sin. 2. 1. _ x. g’(0) = lim. g(h). ___ h. = 0. h  0.

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MY FAVORITE FUNCTIONS

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  1. MY FAVORITE FUNCTIONS OR Continuous from what to WHAT?!?

  2. Section 1.Accordions y = sin x _ x 1 y = sin

  3. f(x) = x sin . 1 x f is continuous at 0 even though there are nearly vertical slopes.

  4. g(x) = x sin 2 1 _ x g’(0) = lim g(h) ___ h = 0 h  0 So g has a derivative at 0. However, there are sequences <a > and <b > such that lim b - a = 0, but lim n n g(b ) - g(a ) ___________ n n = ∞. n n n∞ b - a n∞ n n

  5. Section 2. Le Blancmange function Given x, 0 ≤ x ≤ 1, and non-negative integer n, let k be the greatest non-negative integer such that a = 2-nk ≤ x and b = 2-n(k+1). Define (x,n) (x,n) fn : [0,1][0,1] by fn(x) = min{x-a , b - x}. (x,n) (x,n) k = 0

  6. f(x) = ∑f (x) ¥ ≤ ∑2 ¥ -n n n=1 n=1 Example: 7 7 ; 1 ; 1 . __ 16 __ 16 __ 16 7 __ 16 __ 16 7 __ 16 7 ; __ 16 7 __ 16 f ( ) = f ( ) = f( ) = f ( ) = 1 2 0

  7. Lemma2: Suppose a function h : RR is differentiable at x. If an and if bn are such that n, an ≤ x ≤ bn, then h'(x) = h(bn) - h(an) lim __________ bn - an n∞

  8. Section 3.Stretching zero to one. Cantor's Middle Third SetC is a subset of [0,1] formed inductively by deleting middle third open intervals. Say (1/3,2/3) in step one.

  9. In step two, remove the middle-thirds of the remaining two intervals of step one, they are (1/9,2/9) and (7/9,8/9). In step three, remove the middle thirds of the remaining four intervals. and so on for infinitely many steps.

  10. What we get is C,Cantor’s Middle Thirds Set. C is very “thin” and a “spread out” set whose measure is 0 (since the sum of the lengths of intervals remived from [0,1] is 1. As [0,1] is thick, the following is surprising:

  11. THEOREM. There is a continuous function from C onto [0,1]. The points of C are the points equal to the sums of infinite series of form ∑2s(n)3 ¥ -n where s(n) {0,1}. n=1

  12. F( ∑2s(n)3 ) = ∑s(n)2 ¥ -n ¥ -n n=1 n=1 defines a continuous surjective function whose domain is C and whose range is [0,1]. Example. The two geometric series show F(1/3)=F(2/3)=1/2.

  13. Picturing the proof. Stretch the two halves of step 1 until they join at 1/2. Now stretch the two halves of each pair of step 2 Until they join at 1/4 and 3/4… Each point is moved to the sum of an infinite series.

  14. Section 4.Advancing dimension. 2 NN 2 (2m-1) <m,n> n-1

  15. Theorem. There is a continuous function from [0,1] onto the square. We’ll cheat and do it with the triangle.

  16. Section 5.An addition for the irrationals By an addition for those objects X[0,∞) we mean a continuous function s : X´XX (write x+y instead of s(<x,y>)) such that for x+y the following three rules hold: (1). x+y = y+x (the commutative law) and (2). (x+y)+z = x+(y+z) (the associative law). With sets like Q, the set of positive rationals, the addition inherited from the reals R works, but with the set P of positive irratonals it does not work: (3+√2 )+(3-√2) = 6.

  17. Our aim is to consider another object which has an addition And also “looks like” P. THEOREM5. The set P of positive irrationals has an addition. Continued fraction

  18. Let G(x) denote the greatest integer ≤ x. Let a0 = G(x). Given an irrational x, the sequence <an> is computed as follows: If a0,,,an have been found as below, let an+1 = G(1/r).

  19. Continuing in this fashion we get a sequence which converges to x. Often the result is denoted by However, here we denote it by CF(x) = < a0, a1, a2,a3,…>.

  20. We let <2> denote the constant < 2, 2, 2, 2,…>. Note <2> = CF(1+ √2) since 2 __x 1 or x - 2x - 1 = 0. Hint: A quick way to prove the above is to solve for x in x = 2+

  21. Prove <1> = CF( ) and <1,2> = CF( ) We add two continued fractions “pointwise,” so <1>+<1,2> =<2,3> or <2,3,2,3,2,3,2,3,…>. • Here are the first few terms for , <3,7,15,1,…>. • No wonder your grade school teacher told you • = 3 + . The first four terms of CF(), <3,7,15,1> approximate  to 5 decimals. 1 __ 7 Here are the first few terms fo e, <2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,….>

  22. Lemma. Two irrationals x and y are "close" as real numbers iff the "first few" partial continued fractions of CF(x) and CF(y) are identical. For example <2,2,2,2,2,1,1,1,…> and <2> are close, but <2,2,2,2,2,2,2,2,2,91,5,5,…> and <2> are closer. Here is the “addition:” We define xy = z if CF(z) = CF(x) + CF(y). Then the lemma shows  is continuous.

  23. However, strange things happen:

  24. problems 1. How many derivatives has 1 g(x) = x sin 2 _ x 2. Prove that each number in [0,2] is the sum of two members of the Cantor set.

  25. Problems 3. Prove there is no distance non-increasing function whose domain is a closed interval in N and whose range is the unit square [0,1] ´ [0,1]. 4. Determine

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