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Given a function $f$ with $f(x,y)=\frac{y}{y+x^4}$. Can you extend this function so that it is continuous at $(0,0)$.

I tried different paths and I ended up with different limits (for $y=x$ I ended up with a limit of $1$ and for $y=x^4$ I ended up with a limit of $\frac{1}{2}$. So I believe the answer would be no this function can't be extended so that it is continuous in $(0,0)$ because $f(x,y)$ has different limits on different paths through $(0,0)$. I assume this is correct (please tell me if it isn't).

But how would one go about answering this question if it were possible for let's say a function $g$ at the point $(0,0)$? I would have to define $g(0,0)$, wouldn't I? But how should I define $g(0,0)$?

Peter
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    If the limit $\lim_{(x,y)\rightarrow (0,0)}g(x,y)$ exists, then in order to make the function $g$ continuous, this must be the value you assign to $g(0,0)$. – Roland Apr 20 '16 at 14:50
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    With $y=x^4$ the limit should be $\frac12$. – user296113 Apr 20 '16 at 14:52
  • A counter-example works to show the limit does not exist, but what counter-example would be fruitful depends a lot on the function. A typical approach could involve going for a radial basis change. – mathreadler Apr 20 '16 at 14:53
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    The snag about this function is that it is has the value $-\infty$ for all $x,y$ such that $y=-x^4$. So why would it be continuous at the origin? – almagest Apr 20 '16 at 14:53
  • @Roland, I understand that but how do I find that value before I start proving that it is indeed the limit? – Peter Apr 20 '16 at 14:58
  • @user296113, you're totally right. I was looking at another exercise when writing this down. – Peter Apr 20 '16 at 14:59

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