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If $f:\mathbb R^2\to \mathbb R$ is given by $f(x,y) = \left\{ \begin{array}{cc} [\frac{\sin x}{x}]+[\frac{y}{\sin y}] & \mbox{if } xy \neq 0 \\ 2 & \mbox{if } xy = 0 \end{array} \right.$

Is the function continuous at $(0,0)$. $[.]$ denotes the greatest integer function.

I am on the opinion that the function is discontinuous there at but couldn't find the suitable sequence to show that. Please help!

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Since $\sin x<x$ for $x>0$, we have $0<\frac{\sin(x)}{x}<1$ for $0<x\le\pi$ and therefore

$$\left\lfloor \frac{\sin(x)}{x}\right\rfloor = 0$$

for $x>0$ small.

We also know that $$\frac{x}{\sin x}>1$$

Since $\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$ by L'Hôpital's rule, we conclude that

$$1<\frac{x}{\sin x}<2$$

for small $x>0$. Thus

$$\left\lfloor \frac{x}{\sin x}\right\rfloor = 1$$

for $x>0$ small.

In particular, we can pick $x_n\rightarrow 0+, y_n\rightarrow 0+$ such that

$$f(x_n,y_n)\rightarrow 1\not=2=f(0,0)$$

so $f$ is not continuous (e.g. $x_n=y_n=1/n$).

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