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Math Help - f(x)=sqrt x is continuous

  1. #1
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    f(x)=sqrt x is continuous

    I need help with this proof. I have to use the definition of continuity to prove that the function f defined by f(x)=sqrt x is continuous at every nonnegative real number.
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  2. #2
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    I'll leave it to you to prove that it's continuous at 0 (it's quite easy).

    Suppose x is non-zero and let \varepsilon >0 be given. Then, we seek \delta > 0 such that

    \left|\sqrt{x}-\sqrt{y}\right| < \varepsilon .

    Use the fact that \left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right) = x-y and the assumption that x is nonzero and you should find the desired \delta quite easily.
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