multivariable calculus - Example of discontinuous function satisfying some conditions

I would like to construct a function $f :\mathbb{R}^2 \mapsto \mathbb{R}$ discontinuous at the origin but satisfy the following:

$$\lim_{x\to 0} f(x,mx) = f(0), \quad \lim_{y\to 0} f(my,y)= f(0) \qquad \forall m \in \mathbb{R}$$
That is a function “continuous” along all the lines at a point but still not continuous.

Answer

Let $g\colon S^1\to\Bbb R$ be an unbounded function. Then set
$f(x,y)=rg(\frac xr,\frac yr)$ where $r:=\sqrt{x^2+y^2}$ (and set $f(0,0)=0$). This function is not continuous at the origin, but has the desired property.

---

A bit more explicitly:
Define $h\colon \Bbb R\to\Bbb R$,

$$h(x)=\begin{cases}x+1&x\in\Bbb Q\\x&x\notin\Bbb Q\end{cases}$$

and then
$$f(x,y)=\begin{cases}0&x=0\\\sqrt{x^2+y^2}\cdot h(\frac yx)&x\ne 0\end{cases}$$
is nowhere continuous and has the desired property