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Me and my mates are crunching this question for a while now. While we know that $\sin(xy)$ is continuous , $1 / y $ as the other part of the function clearly has a continuity gap at $y = 0 $, though the function can be continued at $y = 0$ with $f(x,0) = 0 $- why is that? We tried some things but are not getting to the important step that proves the matter.

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Recall that for $z$ real, $|sin (z)| \leq |z|$.

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  • $\begingroup$ So that means: $sin(xy)/y$ $\leq$ $|sin(xy)|/|y|$ $\leq$ $|xy|/|y|$ $=$ $|x|$ ... that means we can ignore y at 0 completley but I guess I am missing something crutial again here ... $\endgroup$
    – salbeira
    Sep 20, 2013 at 12:52
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    $\begingroup$ No, you aren't missing anything - this demonstrates that the limit as $(x,y)\rightarrow (0,0)$ of $sin(xy)/y$ is $0$, meaning if you define the function to be $0$ at $(0,0)$ it's continuous. $\endgroup$
    – Matt Rigby
    Sep 20, 2013 at 12:54
  • $\begingroup$ Right, I confused the conclusion that y is irrelevant in this case, but actually x is irrelevant to remove the singularity, so for any x we can replace f(x,0) with 0 to continue the function. Thanks. $\endgroup$
    – salbeira
    Sep 20, 2013 at 12:58
  • $\begingroup$ No worries. The idea is that due to that inequality for sine, $sin(xy)$ behaves similarly to $xy$ near the origin, so the function as a whole behaves like $x$. $\endgroup$
    – Matt Rigby
    Sep 20, 2013 at 13:01
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You can replace $f(x, 0) = x$. Let's fix $x = a\neq 0$. Then the function essentially becomes $g(y) = \frac{\sin (ay)}{y}$. To figure out the limit for "$g(0)$", we can substitute $z = ay$: $$ \lim_{y \to 0}f(a, y) = \lim_{z \to 0}\frac{\sin z}{z/a} = a\cdot\lim_{z \to 0}\frac{\sin z}{z} = a $$ For $x = 0$, we have that $f(0, y) = 0$ for non-zero $y$, so $f(0, 0) = 0$ is a natural extension at the origin as well.

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I suppose the question is why the function is continuous for $x,y$ in $\mathbb{R}$ ...

The function is obviously continuous in every $(x,y)$, where $x$ and $y$ are non-zero. Let's study the $3$ cases, when $x=0$, $y=0$, and $(x,y)=(0,0)$

1.Let $x$ be any non-zero Real number: $\frac{\sin(xy)}{y} = \frac{\sin(xy)}{xy}x \to 1\times x$ when $y\to0$, so the function is continous and the limit is $x$

  1. Let y be any non-zero Real number: $\frac{\sin(xy)}{y} = 0$ when $x=0$, obvious continuous

  2. When $(x,y)\to(0,0)$, from $1$ and $2$ above, the function has a limit, $0$.

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This seems to be wrong if the question is whether the function $$ f:\mathbb{R}^2 \to \mathbb{R}:(x,y) \mapsto \frac{\sin(xy)}{y}, $$ extended by $0$ where $y=0$ is continuous. Take any point $(a,0)$. Then $(a,1/n)$ is a sequence that converges to this point, so by continuity $f(a,1/n)$ should converge to $f(a,0)=0$. However $$ \frac{\sin(a/n)}{1/n}=\frac{\sin(a/n)}{a/n} \cdot a \to a, $$ so if anywhere, it can only be continuous in the origin.

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The expression $$f(x,y):={\sin(xy)\over y}$$ is at face value undefined when $y=0$, but wait: When $y\ne 0$ one has the identity $${\sin(xy)\over y}=\int_0^x\cos(t\>y)\ dt\ .$$ Here the right side is obviously a continuous function of $x$ and $y$ in all of ${\mathbb R}^2$. It follows that the given $f$ can be extended continuously to the full plane in a unique way.

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