Let $f(x,y)$ be a function of two variables such that $\displaystyle\lim_{(x,y)\to(0,0)}=5$. Which is true? Let $f(x,y)$ be a function of two variables such that $\displaystyle\lim_{(x,y)\to(0,0)}=5$

Which is true?

(a) We have $f(0,0)=5$

(b) If $(x,y)$ approaches $(0,0)$ but not reaching $(0,0)$, then $f(x,y)$ approaches $5$ but not reaching $5.$ 

(c) If $(x,y)$ approaches and reaches $(0,0)$, then $f(x,y)$ approaches and reaches $5$

(d) None of them are true.

My solution


  For (b), this seems to be true as this is essentially the definition of limits, isn't it? As $x$ reaches $0,$ y reaches $0,$ but not at $(0,0)$, $f(x,y)$ reaches $5$, but not at $5$ exactly. 


But my answer says otherwise..

Please enlighten and thanks!
They are all false.

a) and c) Let f(x) =
$\begin{cases}
5, &amp; \text {if (x,y)} \neq \text{(0,0)} \\
0, &amp; \text{otherwise}
\end{cases}$

b) $f(x,y) = 5$
a) false because there could be a discontinuity

b) false because $f(x,y)$ could be equal to $5$ on the entire interval approaching it.  In that case "but not reaching it" disqualifies this choice.  

c) True. The hypothesis says that $f(x,y)$ approaches $5$ as $(x,y)$ approaches $0$.  Since choice c lets $(x,y)$ reach its limit, $f(x,y)$ reaches $5$.