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Math Help - Redefining a function in R2 to be continuous

  1. #1
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    Redefining a function in R2 to be continuous

    The following a continuity question and the answer I put. My professor marked me down, I dont understand his comments, please help.

    Let f(x,y) = sin(xy)/x, for x \neq 0 .
    How would you define f(0,y) for y in R so that f is a continuous function
    on all of R2

    my answer: (*note the back of the book said use f(0,y) = y)
    We have to define f(0,y) so that f(0,y)  \rightarrow 0 as (x,y)  \rightarrow (0,0)

    By a theorem (stated in the book) we already know f(x,y) is a continuous at all points except where x = 0.

    So let f(0,y) =  y^{2}
     y^{2} \rightarrow 0 as (x,y)  \rightarrow 0

    professor's comments: Also trouble at ALL points (0,y). Not just (0,0)



    I dont really see what my professor is talking about...Every single "path" to the orgin must lead to the same result. Moving along the x axis leads us to 0, so moving along the y axis must lead to 0.  y^{2} is clearly a continuous function...and I do not see the difference between the back of the book's answer f(0,y) = y and my answer...

    Please show me what I did wrong. Much appreciated.
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  2. #2
    Member billa's Avatar
    Joined
    Oct 2008
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    Here's a thought - but note that I am not as smart as you!


    I think he is trying to say that there is "trouble" at all points of the function f(0,y), not just the point where y=0. So, I think you should have shown that the function is continuous at all points (0,y), and not just the point (0,0).

    Here is what you did - you showed that the function was continuous for all points (x,y) where x is not 0, and then you showed that the function was continuous at the origin. This leaves many points of the form (0,y) where y is not 0.

    For example, is f continuous at (0,2)?
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