MATH 5070-001: APPLIED ANALYSIS
Fall 1998, University of Colorado 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 M be a metric space with just two elements: a (apple) and o (orange). Let function f(z) be defined on the set of complex numbers C equipped with the standard metric, taking value a if Re(z) is rational, and value o otherwise. Is f(z) continuous at z=(0,0)?
2. Suppose a function f(x) has a right-hand derivative at any point of the interval (0,1). Is it correct that such function must be continuous on (0,1)? Prove, or find a counter-example.
3. Use the definition of the limit in the definition of the derivative to check whether the function | x | (the absolute value of x) is differentiable on (-1,1).
4. Use the sequence-based definition of continuity to check whether the function sign (x) is continuos at some point x on the interval (-1,1). Consider different cases, if necessary.
5. (For extra credit). Let function f(x) be integrable on [a,b]. Is it correct that the function [f(x)], i.e. the integral part of f(x), must be integrable on [a,b]? Prove, or find a counter-example.