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Math Help - continuity

  1. #1
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    continuity

    Use the definition of continuity to prove that the function f defined by f(x) = sqrt(x) is continuous at every nonnegative real number.

    thanks!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Let (a,b) be an open interval in D = [0, \infty. Then

    f^{-1}\left((a,b)\right) = \{x \in D : f(x) \in (a,b)\} = \{x \in D : a < \sqrt x < b\}

     = \{x \in D : a^2 < x < b^2\} = (a^2,b^2).

    Since the inverse image of every open interval is an open interval, f is continuous on D.
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