Results 1 to 2 of 2

Math Help - Multiplication of uniformly continuous functions

  1. #1
    Junior Member
    Joined
    Mar 2011
    Posts
    34

    Multiplication of uniformly continuous functions

    I'm trying to prove:

    If f: A --> R and g: A --> R are uniformly continuous functions, then f * g is uniformly continuous.

    I've gotten to the fact that
    |f(x)g(x) - f(y)g(y)| <= |f(x)| * |g(x)-g(y)| + |g(y)| * |f(x) - f(y)|

    But I'm not sure where to go from here...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by moses View Post
    I'm trying to prove:

    If f: A --> R and g: A --> R are uniformly continuous functions, then f * g is uniformly continuous.

    I've gotten to the fact that
    |f(x)g(x) - f(y)g(y)| <= |f(x)| * |g(x)-g(y)| + |g(y)| * |f(x) - f(y)|

    But I'm not sure where to go from here...
    We really need some more information to solve this problem. We need some information about the set A.

    For example let f(x)=g(x)=x both of these are uniformly continuous on the set [0,\infty) let \delta =\epsilon

    Now consider their prouduct h:[0,\infty) \to \mathbb{R},h(x)=f(x)g(x)=x^2

    on this unbounded set this is not uniformly continuous.

    So if A is compact. The proof is easy! because continuous functions on compact sets are bounded.

    If A is an open interval (a,b) the function is uniformly continuous if and only if it can be continuously extended to the closed compact set [a,b]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Uniformly continuous functions
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 18th 2011, 09:04 AM
  2. Uniformly Continuous f and g functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 12th 2010, 10:02 PM
  3. Multiplying Uniformly Continuous functions
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: March 25th 2010, 06:43 PM
  4. Replies: 3
    Last Post: October 20th 2008, 01:13 PM
  5. Are periodic functions uniformly continuous?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 6th 2008, 08:57 PM

Search Tags


/mathhelpforum @mathhelpforum