Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets Goodi Khalsa, Charles Ouyang, and Hannah Skavdal REU 2012 Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Recap: Definitions Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Given a real-valued function f , the limit of f exists at a point c 2 R if for each given " > 0, there exists δ > 0 such that for any x 2 R if 0 < jc − xj < δ, then jf (c) − Lj < ". Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Given a real-valued function f , f is continuous at a point c 2 R if for each given " > 0, there exists δ > 0 such that for any x 2 R if jc − xj < δ, then jf (c) − f (x)j < ". Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Notation Given f :[a; b] ! R, let FC denote the set of points where f is continuous. Let F+ denote the set of points where the right-sided limit of f exists. Similarly, let F− denote the set of points where the left-sided limit of f exists. Then, also let FL designate the set of points where a one-sided limit exists, that is FL = F+ [ F−. Definition Let B be a subset of R. B is said to be dense in R if for any point x 2 R, x is either in B or a limit point of B. Equivalently, given any " > 0, and any x 2 R, then 9p 2 B such that p 2 N"(x). Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition Let B be a subset of R. B is said to be dense in R if for any point x 2 R, x is either in B or a limit point of B. Equivalently, given any " > 0, and any x 2 R, then 9p 2 B such that p 2 N"(x). Notation Given f :[a; b] ! R, let FC denote the set of points where f is continuous. Let F+ denote the set of points where the right-sided limit of f exists. Similarly, let F− denote the set of points where the left-sided limit of f exists. Then, also let FL designate the set of points where a one-sided limit exists, that is FL = F+ [ F−. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Notation Let D be the set of points of discontinuity of a real function f . Then each point in D belongs to one of the following 3 sets: D1, the set of points c 2 [a; b] such that f has either a removable or jump discontinuity at c. D2, the set of points c 2 [a; b] such that f has only either a right-sided limit or a left-sided limit at c. D3, the set of points c 2 [a; b] such that f has neither a right-sided limit or a left-sided limit at c. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition A set A is countable if there exists a one-to-one correspondence from A to N. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Recap: Results Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Proposition Let f be a function defined on [a; b]. Then the sets D1 and D2 are at most countable. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem Let f be a function defined on [a; b]. Assume that the set FL is dense in [a; b]. Then the set FC is nonempty, dense in [a; b], and uncountable. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem Let f :[a; b] ! R, with FL dense in R. Then f is unique on FC . Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Corollary If f :[a; b] ! R and m(D3) = 0, and given FL dense in R, then f is unique except at a set of measure zero. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Proposition Given a function f :[a; b] ! R, there is no countable dense set G where G := fc : lim f (x) = L with L 2 g. x!c R Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets The set of continuity points of a function f : R ! R is a Gδ set. Conversely, every Gδ subset of R is the set of continuity points of a function f : R ! R. Definition An Fσ set is a countable union of closed sets. A Gδ set is a countable intersection of open sets. Examples: Q is Fσ. RnQ is Gδ. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition An Fσ set is a countable union of closed sets. A Gδ set is a countable intersection of open sets. Examples: Q is Fσ. RnQ is Gδ. The set of continuity points of a function f : R ! R is a Gδ set. Conversely, every Gδ subset of R is the set of continuity points of a function f : R ! R. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets 1 Er = fa : !(f ; a) ≥ r g Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem c (FL) is an Fσ set. Let !(f ; J) = sup f (x) − inf f (x) x2J x2J !(f ; a) = inf (!(f ; J)) a2J 1 Er = fa : !(f ; a) ≥ r g Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Theorem There is no function that is continuous only on Q. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition A real number x is a Liouville number if for any given n 2 N, there exist infinitely many relatively prime integers p and q with q > 1 p 1 such that 0 < jx − q j < qn . We will denote the set of Liouville numbers with L. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Let f : R ! R 0; if x 2 L [ f (x) = Q 1 ; otherwise Nx Where Nx is defined as the first n where x fails to meet the definition of a Liouville number. In other words, the first n such that there do not exist p and q such that p 1 0 < jx − q j < qn . Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Is there a Gδ set of measure zero, containing Q, that we can construct a continuous function on? Let f : R ! R 0; if x 2 L [ f (x) = Q 1 ; otherwise Nx Where Nx is defined as the first n where x fails to meet the definition of a Liouville number. In other words, the first n such that there do not exist p and q such that p 1 0 < jx − q j < qn . Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition The irrationality measure for a real number x is a numeric representation of how well x can be approximated by the rationals. p 1 Let µ be the least upper bound such that 0 < jx − q j < qµ where p; q 2 Z. We call µ the irrationality measure of x. Notation Let µ(x) stand for irrationality measure of x. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets Definition If r is a root of a nonzero polynomial equation n n−1 anx + an−1x + ··· + a1x + a0 = 0 where ai 2 Z and r satisfies no similar equation of degree < n then r is said to be an algebraic number of degree n. A number that is not algebraic is said to be transcendental. For example: µ(x) = 1 if x 2 L, µ(x) = 1 if x 2 Q, µ(x) = 2 if x is an algebraic number of degree > 1, µ(x) ≥ 2 if x is transcendental. Goodi Khalsa, Charles Ouyang, and Hannah Skavdal Construction of Continuous Functions on Special Sets For example: µ(x) = 1 if x 2 L, µ(x) = 1 if x 2 Q, µ(x) = 2 if x is an algebraic number of degree > 1, µ(x) ≥ 2 if x is transcendental.