Continuity of $f(x,y)$

Question

So, I need to prove that the function-
$$f(x,y)=
\begin{cases}
\dfrac{x^2+y^2}{x-y} & (x,y)\neq (0,0) \\
0 & (x,y) =(0,0)\\
\end{cases}$$
is discontinuous at the origin. But to me it seems like this is continuous there. I tried solving this in the following manner.
Suppose we approach the origin along the line $y=mx$, thus $$\lim_{(x,y) \to (0,0)} f(x,y)=\lim_{x \to 0} \dfrac{x(1+m^2)}{1-m}$$
Isn't the above expression $=0$ for all values of $m$. Then why is this function discontinuous at the origin as my textbook says?

Answer 1

Note that$$f(y+y^3,y)=y^3+2y+\frac2y$$and that therefore$$\lim_{y\to0^+}f(y+y^3,y)=\infty.$$