Results 1 to 2 of 2

Math Help - Continuous Functions

  1. #1
    Newbie
    Joined
    Nov 2010
    Posts
    12

    Continuous Functions

    I have difficulties solving the following exercise:

    Prove that if f and g are continuous functions from a topological space X to \mathbb{R}, then f+g and fg are continuous.

    The hint says: Apply this theorem: let X_0, X_1 and X_2 be topological spaces and let f:X_0 \Rightarrow X_1 and g: X_1 \Rightarrow X_2 be continuous functions. Then g(f): X_0 \Rightarrow X_2 is continuous.

    and use the easy facts that the maps (x,y) to x+y and xy are continuous (where defined).
    I really don't know how to use the hint, i can't imagine that this exercise is difficult.

    Thanks for help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    Okay, they've given you that the maps (x,y)\mapsto xy and (x,y)\mapsto x+y are continuous maps from \mathbb{R}\times\mathbb{R}\to\mathbb{R}. A property of the product topology is that a composite function is continuous if and only if its component functions are. Use the fact that a composition of continuous functions is continuous and you're done.

    Edit: That might not be clear. The point is that the map (f+g):X\times X\to\mathbb{R} is a composition of two maps, F:X\times X\to\mathbb{R}\times\mathbb{R} where F(x_1,x_2)=(f(x_1),g(x_2)) and s:\mathbb{R}\times\mathbb{R}\to\mathbb{R}, where the "s" stands for "sum", and s(x,y)=x+y. They've told you s is continuous, and you know F is continuous since its components are continuous and its codomain has the product topology. Therefore the composition s(F(x,y)) is continuous.
    Last edited by Tinyboss; January 22nd 2011 at 11:16 AM. Reason: clarify
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Continuous Functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 29th 2010, 03:44 PM
  2. Continuous Functions
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 26th 2010, 09:52 PM
  3. more continuous functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 5th 2009, 12:57 PM
  4. continuous functions
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 4th 2009, 01:47 PM
  5. Replies: 1
    Last Post: January 27th 2008, 09:03 PM

/mathhelpforum @mathhelpforum