Let function $\,\, f:\mathbb{R}^2\rightarrow\mathbb{R}\,$ be defined as
$$
f(x,y) =
\begin{cases}
\frac{x^ny^m}{x^{n'}+y^{m'}}\text{,} &\quad(x,y)\neq(0,0)\\
\text{0,} &\quad \text{otherwise}\\
\end{cases}
$$
with $n,m,n',m' \in\mathbb{N}^*$ so that $n>n'$ or $m>m' $.
Then from previous examples I've analysed, I'd guess that $\,f$ is continous at $(0,0)$. But somehow I've got a feeling that this can't be true. (My feeling is caused by the sequence criterion. Suppose $n>n'$ and $m'>m$. If $y$ "shrinks a lot faster" in a given sequence, $\,f(x,y)$ should grow instead of converging to 0.)
If $\,\boldsymbol{f}$ is continus at origin could you either:
- proof it
- explain your intuition why this is true
If $\,\boldsymbol{f}$ is not continus at origin could you either:
- hint me a counter-example
- disproof the statement
Cheers,
Pascal