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Math Help - Continuity with product of the range

  1. #1
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    Continuity with product of the range

    Let f_1: X \rightarrow Y_1 and f_2: X \rightarrow Y_2 be continuous functions. Show that h: X \rightarrow Y_1\times Y_2 defined by h(x)=(f_1(x),f_2(x)), is continuous as well.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Let f_1: X \rightarrow Y_1 and f_2: X \rightarrow Y_2 be continuous functions. Show that h: X \rightarrow Y_1\times Y_2 defined by h(x)=(f_1(x),f_2(x)), is continuous as well.
    Let W be a neighborhood of h(x), x \in X such that W = U \times V where U is a neighborhood of f_1(x) and V is a neighborhood of f_2(x).
    Let p be a point in X that belongs to h^{-1}(U \times V). Then, h(p) \in U \times V iff f_1(p) \in U and  f_2(p) \in V. Thus, h^{-1}(W) = h^{-1}(U \times V) = f_1^{-1}(U) \cap f_2^{-1}(V). Since f_1 and f_2 are continuous and an intersection of open sets is open, h^{-1}(W) is open. Thus, h is continuous.
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  3. #3
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    Quote Originally Posted by Andreamet View Post
    Let f_1: X \rightarrow Y_1 and f_2: X \rightarrow Y_2 be continuous functions. Show that h: X \rightarrow Y_1\times Y_2 defined by h(x)=(f_1(x),f_2(x)), is continuous as well.
    Let ε>o and aεX.

    Since \lim_{x\rightarrow a}{f_{1}(x)}=f_{1}(a) and

    \lim_{x\rightarrow a}{f_{2}(x)} =f_{2}(a),then there exist:

    \delta_{1}>0 and such that:

    if |x-a|<\delta_{1} and xεX ,then |f_{1}(x)-f_{1}(a)|<ε/2 for all,x............................................. .......................................1

    \delta_{2}>0 and such that:

    if |x-a|<\delta_{2} and xεX, then |f_{2}(x)-f_{2}(a)|<ε/2 for all ,x................................................ ...........2.


    Choose \delta = min{ \delta_{1},\delta_{2}}


    Let |x-a|<δ and xεX.

    then |x-a|<\delta_{1} and |x-a|<\delta_{2} and by (1) and (2) we have:


    |f_{1}(x)-f_{1}(a)|+|f_{2}(x)-f_{2}(a)|<\epsilon


    BUT.

    Norm (h(x)-h(a)) = ||h(x)-h(a)|| = \sqrt{(f_{1}(x)-f_{1}(a))^2 + (f_{2}(x)-f_{2}(a))^2}\leq|f_{1}(x)-f_{1}(a)| + |f_{2}(x)-f_{2}(a)|<\epsilon

    Thus \lim_{x\rightarrow a}h(x) = h(a),for all ,a in X AND hence the function ,h is continuous over X
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