Consider the function $f$ whose graph is shown below.

This function is given by $$f(x,y) = \begin{cases} \frac{2xy}{x^{2}+y^{2}}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0). \end{cases}$$

(a) Find a formula for the single variable function $f(0,y)$.
$f(0,y) =$
What is $f(0,0)$ for this function?
$f(0,0) =$
Find its limit as $y\to 0$:
$\lim\limits_{y\to 0} f(0,y) =$

(b) Based on your work in (a), is the single variable function $f(0,y)$ continuous? ? yes no

(c) Next, similarly consider $f(x,0)$.
$f(x,0) =$

$f(0,0) =$
$\lim\limits_{x\to 0} f(x,0) =$

(d) Based on this work in (a), is the single variable function $f(x,0)$ continuous? ? yes no

(e) Finally, consider $f$ along rays emanating from the origin. Note that these are given by $y = m x$, for some (constant) value of $m$.

Find and simplify $f$ on the ray $y = x$:
$f(x,x) =$

(Notice that this means that $y=x$ is a contour of $f$. Be sure you can explain why this is.)

Find and simplify $f$ on any ray $y = m x$.
$f(x,mx) =$

(Again, notice that this means that any ray $y=mx$ is a contour of $f$; be sure you can explain why.)

(f) Is $f(x,y)$ continuous at $(0,0)$? ? yes no