To demonstrate that the product of two uniformly continuous functions, f and g, is uniformly continuous on a bounded set X, both functions must be uniformly continuous and bounded. The proof involves showing that the expression |f(x)g(x) - f(y)g(y)| can be made arbitrarily small by controlling |x - y| through the use of epsilon-delta definitions. The hint provided emphasizes breaking down the product into manageable parts, leveraging the uniform continuity of f and g to establish the desired result.
PREREQUISITESMathematics students, particularly those studying real analysis, educators teaching continuity concepts, and anyone interested in the properties of functions in bounded domains.
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