Suppose we have a bounded continuous function $f(x)$ on some interval $(a,b)$. Suppose we also have an function $g(x)$ that is uniformly continuous on the same interval $(a,b)$. Then, is the product $f(x)g(x)$ uniformly continuous? Intuition tells me that it is true. However, I am not sure. Most of the cases that I've dealt with involve relating an explicit formula for the function (the domain) to an epsilon and a delta. I have tried using the same approach as done in product of two uniformly continuous functions is uniformly continuous
However, I am stuck at the same part. I do not know how to bound them.
Best Answer
This is not true.
Let $g(x)=1$, the interval $I$ be $(0,1)$ and $f(x)=\sin\left(\dfrac{1}{x}\right)$.
Then applying the theorem would imply that $f(x)=\sin\left(\dfrac{1}{x}\right)$ is uniformly continuous on $(0,1)$ which is incorrect(See here for proof).