Homework Assignment 6 For Math 104 Lecture 002 Summer 2014

Homework Assignment 6 For Math 104 Lecture 002 Summer 2014jason Fergu

Prove that the function f, defined on the real numbers as follows: if x is irrational, then f(x) = 0; if x is rational and written in lowest terms as p/q, then f(x) = 1/q, is continuous at every irrational number and discontinuous at every rational number.

Let M be any metric space, and A and B any subsets of M. Prove:

• a. The interior of A equals the complement of the closure of the complement of A: int A = M \ (M \ A).
• b. The union of A and B: A ∪ B = A ∪ B.
• c. A is closed if and only if A equals its own closure.
• d. A equals its own interior.
• e. If A is a subset of B: A ⊆ B.
• f. Describe the analogous statements replacing closure with interior.

For a point p in a metric space M and > 0:

• a. Show that the set of points within radius r from p: M(p) ⊆ {q ∈ M : d(p,q) ≤ r}.
• b. In R², using Euclidean distance, show that the same set is exactly the points within radius r: M(p) = {q ∈ M : d(p,q) ≤ r}. Generalize to Rⁿ and normed vector spaces.
• c. Provide an example where the set of points within radius r differs from the closed ball: M(p) ≠ {q ∈ M : d(p,q) ≤ r}.

Prove that the restriction of a continuous function to a subset with the subspace topology remains continuous.

Show that defining functions piecewise by gluing together continuous functions on overlapping subspaces yields a continuous global function, given suitable agreement conditions.

Given a metric space M, a cover of M by open or closed sets, and continuous functions defined on these subsets that agree on overlaps, prove that the global functions formed by patching the local functions are continuous.

Prove that the set S = { x ∈ Q : -√2

For sequences (xₙ) and (yₙ) in M converging to x and y, respectively, show that the sequence of distances d(xₙ, yₙ) converges to d(x, y), and explain why d: M × M → R is continuous.

Reflect on your time spent on this assignment, its difficulty, and appropriateness of the questions.

Paper For Above instruction

The function f, defined on the real line with the specified properties, exhibits interesting behavior regarding continuity at rational and irrational points. This function, often referred to as the rational ruler or Thomae's function, takes on a value of 0 at irrationals and 1/q at rationals when written in lowest form. To investigate its continuity, one considers the definition of limits and the topology of the real line.

At irrational points, for any ε > 0, there exists a δ > 0 such that within the δ-neighborhood, the function values are arbitrarily close to zero. Since irrationals are dense, and the function value at irrationals is always zero, the limit of f(x) as x approaches an irrational point c can be shown to be 0, matching the function value at c. Thus, f is continuous at every irrational number.

Conversely, at a rational point p = p/q, the function values near p include irrationals where the function approaches 0, and rationals approaching p/q where the function takes values 1/r, which can be made arbitrarily small but never fully equal to the function value at p unless q=1. The discontinuity arises because the limit does not equal the function value at rationals with q > 1, leading to discontinuities at every rational point.

In the context of topological spaces, the concepts of closure, interior, and boundary help formalize these ideas. The closure of a set involves including all limit points, and the interior comprises points around which there exists an open neighborhood entirely within the set. Notably, the complement operation interrelates with these properties, and set equalities involving closure and interior provide foundational understanding of topological openness and closedness.

Specifically, for subsets A in a metric space M, the interior is the complement of the closure of the complement, reflecting the duality of open and closed sets. The union and intersection properties of closures and interiors reinforce how these operations behave under set inclusion and set operations, forming a vital part of topology foundations.

Considering metric spaces such as R² with Euclidean distance, the concept of a metric ball M(p) describes all points within radius r. When employing standard Euclidean distance in R², these balls are exactly the sets {q | d(p, q) ≤ r}, a fact that extends to Rⁿ and general normed vector spaces. An explicit example where the set differs from the closed ball illustrates the subtlety in different metric structures or when considering open versus closed balls.

Continuity of functions restricted to subspaces follows naturally from the properties of the subspace topology—the restriction of a continuous function remains continuous because the inverse image of open sets in the codomain is still open in the subspace. This foundational result is critical when analyzing locally defined functions and their compatibility on overlapping regions.

The process of gluing functions involves defining a global function piecewise from local continuous functions on a cover of the domain, ensuring they agree on overlaps. When the cover consists of open sets, the continuity of the global function follows from the compatibility of local functions and their agreement on overlaps. When the cover is finite and comprised of closed sets, similar results hold under suitable conditions, leveraging finiteness and the properties of the subspace topology.

Regarding the set S = { x ∈ Q : -√2

The convergence of sequences (xₙ) and (yₙ) in a metric space M, approaching x and y respectively, guarantees that the sequence d(xₙ, yₙ) converges to d(x, y). This result emphasizes the continuity of the metric function d: M × M → R, since it preserves limits under sequence convergence. These properties underpin many core aspects of metric space topology, illustrating the stability of limits and distances under convergence.

Reflecting on the effort spent on the assignment, its length, and difficulty reveals insights into the comprehension of metric and topological concepts. The questions crafted are designed to challenge understanding of continuity, set operations, and the structure of metric spaces, requiring thorough reasoning and application of foundational theories in analysis and topology.

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