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Math Help - continuous metric spaces

  1. #1
    Member Goku's Avatar
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    continuous metric spaces

    Show that

    1. the projection map p : R^2 → R given by p(x; y) := x is continuous.
    .

    My proofs:

    We want to show that \forall \epsilon > 0, \exists \delta>0 such that:

    d(x,y)< \delta \Rightarrow d(f(x),f(y)) < \epsilon

    (x1,y1) (x2,y2) are points in R^2

    1) d(x1,x2) < \epsilon
    |x1-x2| < \epsilon

    We can define \delta = the square root of ( \epsilon ^2 - (y2-y1)^2)

    Since there exists a delta for any epsilon we choose, implies that p is continuous.

    Is this correct...
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  2. #2
    CSM
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    Looks quite correct to me.
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