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Math Help - epsilon delta proof

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    epsilon delta proof

    Give a rigorus epsilon-delta proof that the function f: R^3 into R defined by f(x,y,z) = yx^2 + 2xz^2 is continuous at (1,1,1).
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    Quote Originally Posted by jburks100 View Post
    Give a rigorus epsilon-delta proof that the function f: R^3 into R defined by f(x,y,z) = yx^2 + 2xz^2 is continuous at (1,1,1).
    suppose \epsilon > 0 is given and choose \delta = \min \{1, \epsilon/21 \}. now if 0 < |x-1|, |y-1|, |z-1| < \delta, then 0 <x,y,z < 2 because \delta \leq 1.

    we also have: yx^2+2xz^2 - 3 = (y-1)x^2 + (x-1)(x+1+2z^2) + 2(z-1)(z+1). thus: |yx^2+2xz^2-3| < 4 \delta + 11 \delta + 6 \delta = 21 \delta < \epsilon.
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