any example of product of two uniformly continuous functions

• Oct 20th 2008, 12:43 PM
szpengchao
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.
• Oct 20th 2008, 12:50 PM
Laurent
Quote:

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.
• Oct 20th 2008, 12:51 PM
szpengchao
wrong
Quote:

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
• Oct 20th 2008, 01:13 PM
Laurent
Quote:

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.