The problem involves demonstrating that the product of two uniformly continuous and bounded functions, f and g, is also uniformly continuous. The context is rooted in the properties of uniform continuity and boundedness within a specified domain X.
The discussion is ongoing, with participants exploring different aspects of the problem. Some have raised questions about the necessity of both functions being bounded and how that relates to uniform continuity. A hint has been provided regarding the use of epsilon-delta arguments to approach the proof.
There is a focus on the definitions of uniform continuity and boundedness, with participants questioning the assumptions and implications of these properties in the context of the problem.
No, you don't "know" that. In fact, here, you are told that they are both uniformly continuous.CarmineCortez said:Homework Statement
suppose f and g are uniformly continuous functions on X
and f and g are bounded on X, show f*g is uniformly continuous.
The Attempt at a Solution
I know that if they are not bounded then they may not be uniformly continuous. ie x^2
and also if only one is bounded they are not necessarily uniformly continuous.
Are you saying you could do this if only one were bounded? Do you understand what it is you are asked to prove?not sure what to do if they are both bounded