MATH 5070-001: APPLIED ANALYSIS
Fall 1997, 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. Let P be another direction such that . Does the function f(x) have a limit in the direction P? Prove, or provide a counterexample.
2. Let function f(z) be defined on a set of complex numbers C, taking values in a metric space of vectors with 2 real components (x,y) equipped with the standard euclidean metric. Namely, let us consider f(z) = (|1/z|,|(1+z)/(2+z)|). Let S be the following direction:

   

   where r is any positive real number. Does the function f(x) satisfy the Cauchy convergence criterion in the direction S?

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

   

   Does it converge uniformly on Q when ?