$\lim_{z\to 0}f(z)=0$ along $y=mx^n$ but for which $\lim_{z\to 0}f(z)$ does not exist Given a nonzero real number $m$, construct a function $f(z)$ such that 
$$
\lim_{z\to 0}f(z) = 0
$$ 
along each curve of the form $y = m\cdot x^n$ (for $n = 1,2,3,4,\dots$), but for which
$$\lim_{z\to 0}f(z)$$
does not exist. I don't know from where I should start.
Thank you ...

$f(z) = 0$ on the union of the curves $y =mx^n$, $f(z) = 1$ in the complementary.
The union of curves $y = mx^n$ with $n\in\Bbb N$ is a "small" set. In particular, the complementary intersects every neighborhood of 0.
If also $m$ is variable you can take
$f(x,y) = 0$ when $|y|\ge e^{-1/x^2}$ or $y=0$ and 1 in the complementary.
