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Math Help - Continuity of fog and f/g on standard topology.

  1. #1
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    Continuity of fog and f/g on standard topology.

    Let $f: X \rightarrow \mathbb{R}$ be continuous functions, where ($X, \tau$) is a topological space and $\mathbb{R}$ is given the standard topology.

    a)Show that the function $f \cdot g : X \rightarrow \mathbb{R}$,defined by\\
    $(f \cdot g)(x) = f(x)g(x)$\\
    is continuous.\\

    b)Let $h: X \setminus \{x \in X | g(x) = 0\}\rightarrow \mathbb{R}$ be defined by\\
    $h(x) = \frac{f(x)}{g(x)}$.\\
    Show that $h$ is continuous.\\

    PICTURE ATTACHED!

    Help!!!
    Attached Thumbnails Attached Thumbnails Continuity of fog and f/g on standard topology.-4.png  
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  2. #2
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    Re: Continuity of fog and f/g on standard topology.

    For (a), think about

    \text{1) } X \rightarrow X \times X, \ x \mapsto (x,x).

    \text{2) } X \times X \rightarrow \mathbb{R} \times \mathbb{R}, (x,y) \ \mapsto (f(x), g(y)).

    \text{3) } \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, (s,t) \ \mapsto st.
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  3. #3
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    Re: Continuity of fog and f/g on standard topology.

    got it. and b) is just the same only h(x) = f(x)k(x) , where k(x) = 1/g(x) , both f and k- continuous?
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  4. #4
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    Re: Continuity of fog and f/g on standard topology.

    Yes, except it's now everywhere on that subspace Y of X where g isn't 0. So everything as above, except with Y instead of X.

    (Did you prove that the diagonal map X -> X x X is continuous? It's a brief but worthwhile exercise.)
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