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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.