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Thread: Continuous Functions

  1. #1
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    Continuous Functions

    I have difficulties solving the following exercise:

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

    The hint says: Apply this theorem: let $\displaystyle X_0$, $\displaystyle X_1$ and $\displaystyle X_2$ be topological spaces and let $\displaystyle f:X_0 \Rightarrow X_1$ and $\displaystyle g: X_1 \Rightarrow X_2$ be continuous functions. Then g(f):$\displaystyle X_0 \Rightarrow X_2$ 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.
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  2. #2
    Senior Member Tinyboss's Avatar
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    Okay, they've given you that the maps $\displaystyle (x,y)\mapsto xy$ and $\displaystyle (x,y)\mapsto x+y$ are continuous maps from $\displaystyle \mathbb{R}\times\mathbb{R}\to\mathbb{R}$. 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 $\displaystyle (f+g):X\times X\to\mathbb{R}$ is a composition of two maps, $\displaystyle F:X\times X\to\mathbb{R}\times\mathbb{R}$ where $\displaystyle F(x_1,x_2)=(f(x_1),g(x_2))$ and $\displaystyle s:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, where the "s" stands for "sum", and $\displaystyle s(x,y)=x+y$. 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 $\displaystyle s(F(x,y))$ is continuous.
    Last edited by Tinyboss; Jan 22nd 2011 at 10:16 AM. Reason: clarify
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