Show that the function $f = \frac{xy}{x^2 + y^2}$ is continuous along every horizontal and every vertical line Consider the function $ f:\mathbb{R^2} \rightarrow \mathbb{R}$ given by $ f(x,y) = \left\\{ \begin{array}{l l} \frac{xy}{x^2 + y^2} & \quad \text{if (x,y) $\neq$ (0,0)}\\\ 0 & \quad \text{if (x,y) = (0,0)} \end{array} \right.$ Show that f is continuous along every horizontal and every vertical line (i.e. for every $x_0, y_0 \epsilon \mathbb{R}$, the functions $g,h:\mathbb{R}\rightarrow \mathbb{R}$ given by $g(t) = f(x_0,t)$ and $h(t)=f(t,y_0)$ are continuous). I know that this function is not continuous at $(0,0)$, but the horizontal/vertical line issue is proving difficult to work with. Any help/hints appreciated!

Along a fixed horizontal line $y = c$ (constant). Put $y = c$ in the function and get $f(x,c) = \frac{xc}{x^2 + c^2}$ when $x \
eq 0$ , which is a function of $x$ only and see it is a continuous one variable function. Do the same for a verticle line.