ECON6012/ECON2125: Semester Two, 2024
Tutorial 5
A Note on Sources
These questions and answers do not originate with me. They have either been influenced by, or directly drawn from, other sources.
Key Concepts
Mappings, Functions, Correspondences, Domain, Co-Domain, Range (Im-age Set), Continuity of Functions, Uniform. Continuity of Functions, Lips-chitz Continuity of Functions, Compactness, Convergent Sequences, Cauchy Sequences, Convergent Subsequences.
Tutorial Questions
Tutorial Question 1
Prove that the function f : R −→ R defined by f (x) = x2
is continuous.
Tutorial Question 2
Prove that the function f : R −→ R defined by f (x) = x2
is not uniformly continuous.
Tutorial Question 3
Prove that the function f : (0, 1) −→ R defined by f (x) = x2
is uniformly continuous.
Tutorial Question 4
Let (X, d) and (Y, r) be metric spaces. Prove that if X is a compact set, then all continuous functions f : X −→ Y are uniformly continuous.
Additional Practice Questions
Additional Practice Question 1
Prove that the function f : (0,∞) −→ R defined by f (x) = x/1
is continuous.
Additional Practice Question 2
Prove that the function f : (0,∞) −→ R defined by f (x) = x/1
is not uniformly continuous.