Why is $\sin(xy)/y$ continuous? 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.

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.
