QUESTION:
$$f(x,y)=\begin{cases}x \sin \frac{1}{y} + y \sin \frac{1}{x} &
\text{if } xy \not = 0 \\ 0 & \text{if } xy = 0\end{cases}$$Show that $f(x,y)$ is continuous at $(0,0)$.
MY ATTEMPT:
What I have to do is:
- Show that $f_x$ and $f_y$ both exist at $(0,0)$ and
- Show that either of $f_x$ or $f_y$ is bounded .
Now, first of all, we have that
$$f_x(0,0)=\lim_{h\to0} \frac{f(h,0)-f(0,0)}{h}=0$$ and
$$f_y(0,0)=\lim_{k\to0} \frac{f(0,k)-f(0,0)}{k}=0$$
So the condition$(1)$ is satisfied.
For $(2)$, we have that
$$f_x=\sin \frac{1}{y} – \frac{y}{x^2} \cos \frac{1}{x}$$ and
$$f_y=\sin \frac{1}{x} – \frac{x}{y^2} \cos \frac{1}{y}$$
But then how do I show that $f_x$ or $f_y$ is bounded?
If possible, can you suggest some other method of proof by $\epsilon-\delta$ definition of continuity?
Please help.
Best Answer
Why not directly show the limit is zero?
$$\left|x\sin\frac1y+y\sin\frac1x\right|\le |x|+|y|\xrightarrow[(x,y)\to(0,0)]{}0$$
and we've finished since $\;f(0,0)=0\;$ .