Uniform Continuity of Multivariable Function

Question

Suppose we define a function $f:\mathbb{R}^2\backslash \{0,0\} \rightarrow \mathbb{R}$, with $f(x,y)$ defined as $\frac{xy^3}{x^4+y^4},(x,y) \neq (0,0)$. Now I'm trying to figure out if this function is uniformly continuous. I don't think it is uniformly continuous for the following reason: If we consider $(x,y) \rightarrow (0,0)$ along the path $y=mx$. We get $\underset{(x,y) \rightarrow (0,0)}{lim} f(x,y)=\underset{x\rightarrow 0}{lim} \frac{mx^4}{x^4+mx^4}=\underset{x \rightarrow 0}{lim} \frac{m^4}{1+m^4}$. For different values of $m$, we get different values, so $f(x,y)$ is not continuous at $(0,0)$ and hence $f(x,y)$ cannot be uniformly continuous. Is this thinking correct?

Answer 1

You are right to observe thet the function is not continuous at $(0,0)$. This implies that it is not uniformly continuous in any set containing $(0,0)$. But it is uniformly continuous in any closed and bounded set not containing $(0,0)$. Uniform continuity requires the specification of a set where it holds or not holds.