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Math Help - Proof: f(x,y) Differentiable then it is Continuous

  1. #1
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    Proof: f(x,y) Differentiable then it is Continuous

    Sorry for such a lame question. I would like to know how to do this because it is likely to come up on an exam.

    Prove that if f(x,y) is differentiable at (a,b) then it is continuous at (a,b).

    I was never good at proving things. If I could even get a tip it would be really helpful. Thanks.
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  2. #2
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    Quote Originally Posted by Shananay View Post
    Sorry for such a lame question. I would like to know how to do this because it is likely to come up on an exam.

    Prove that if f(x,y) is differentiable at (a,b) then it is continuous at (a,b).

    I was never good at proving things. If I could even get a tip it would be really helpful. Thanks.
    Always start from definitions and always state the definitions exactly.

    "Differentiable", at (a, b), for a real valued function of two variables, means, there exist numbers A and B and a function \epsilon(x,y) such that
    f(x,y)= f(a,b)+ A(x-a)+ B(x- b)+ \epsilon(x,y) and such that \lim_{(x,y)\to (a,b)} \frac{\epsilon(x,y)}{\sqrt{(x-a)^2+ (y- b)^2}}= 0
    (Essentially that says that f(a,b)+ A(x- a)+ B(x- b) is the "best" linear approximation to f(x,y) around (a, b).)_

    Now, take the limit of both sides of f(x,y)= f(a,b)+ A(x-a)+ B(x- b)+ \epsilon(x,y) as (x, y) goes to (a, b). That condition on \epsilon tells you that it must go to 0.
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