MATH 5070-001: APPLIED ANALYSIS
Fall 1996, University of Colorado at Denver
Instructor: Prof. Andrew V. Knyazev

Test 2

Please, provide complete and detailed proofs of every statement, or give a reference to the book if you use a statement from the book.

1. Let S be a direction on a set E. Let a function f(x) be defined on E, taking values in a metric space equipped with a distance. Suppose that the function f(x) has a limit in the direction S. Let G be such a subset in E, that produces a direction GS. Prove that the function f(x) is bounded in the direction GS.
2. Let function f(k) be defined on a set of natural numbers N, taking values in a metric space of vectors with 2 real components (y,z) equipped with a standard euclidean metric. Namely, we define f(k) = (1/k,(1+k)/(2+k)). Let S be a direction on N:

   

   Does the function f(k) satisfies the Cauchy convergence criterion in the direction S?

3. Let M be a metric space with just two elements: a (apple) and o (orange). Let function f(x) be defined on the set of real numbers R equipped with a standard metric, taking value a if x is rational, and value o otherwise. Is f(x) continuous at x=0?
4. For every natural n let be defined on a set of rational numbers Q equipped with a standard metric, taking values in a set of real numbers R equipped with a standard metric. Namely, let

   

   Does it converge uniformly on Q when ?