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Let \begin{equation} f: \mathbb{R}^{2} \rightarrow \mathbb{R} \end{equation} be a function of two real variables given by \begin{equation} f(x,y) = \begin{cases} \frac{x}{y} & \text{for}\quad y\neq 0 \\ 0 & \text{for}\quad y=0 \ \\ \end{cases} \end{equation}

Does $f(0,0)=0$? My understating is that $f(0,0) =0$ since $f(x,0)=0$ for all $x \in \mathbb{R}$. Is this correct?

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    $\begingroup$ Yes, you are right. We have $f(0,0)=0.$ $\endgroup$
    – Fred
    Commented Oct 18, 2022 at 10:55
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    $\begingroup$ Here , $f(0,0) = 0$ , but the limit , $\lim {x,y \rightarrow 0} [ f(x,y) ]$ is "something" else ! $\endgroup$
    – Prem
    Commented Oct 18, 2022 at 11:26

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