# Thread: any example of product of two uniformly continuous functions

1. ## any example of product of two uniformly continuous functions

f,g are uniformly continuous on [a,b]

any example f*g which is not uniformly continuous.
and :

example, f,g are bounded but fg is uniformly continuous.

2. Originally Posted by szpengchao
f,g are uniformly continuous on [a,b]

any example f*g which is not uniformly continuous.
and :

example, f,g are bounded but fg is uniformly continuous.
Is this what you meant to write?
Anyway, if you're looking for functions $\displaystyle f,g$ on $\displaystyle [a,b]$ that are uniformly continuous and such that their product is not, you won't find any. Indeed, on a segment, uniformly continuous is equivalent to continuous (this is called Heine Theorem (at least in France)), and the product of two continuous functions is continuous.

3. ## wrong

Originally Posted by Laurent
Is this what you meant to write?
If you look for functions $\displaystyle f,g$ on $\displaystyle [a,b]$ that are uniformly continuous and such that their product is not, you won't find any. Indeed, on a segment, uniformly continuous is equivalent to continuous, and the product of two continuous functions is continuous. This is called Heine Theorem (at least in France).

sorry.made a mistake. on subset X in R

4. Originally Posted by szpengchao
sorry.made a mistake. on subset X in R
Check that $\displaystyle x\mapsto x$ is uniformly continuous on $\displaystyle \mathbb{R}$, while $\displaystyle x\mapsto x^2$ is not.

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### product of uniform continous function need not be uniform continous why

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