# Math Help - Uniformly continuous function-two variables

1. ## Uniformly continuous function-two variables

The question:

Given $f(x,y)=\cos(\frac{1}{1-x^2-y^2})$, prove:

a) $f(x,y)$ isn't uniformly continuous functions on $D=\{(x,y)|x^2+y^2<1\}$

b) $f(x,y)$ is uniformly continuous functions on $D=\{(x,y)| 3 \leqslant x^2+y^2\leqslant 4\}$

a) First I say,

$x^2+y^2=t$, $0 \leqslant t<1$

now,

$f(x,y)$ becomes to be function with one variable- $t$:

$f(t)=\cos(\frac{1}{1-t}$.

Now I pick two points $t_0,t_1 \in [0,1)$:

$t_0=1-\frac{1}{2\pi n}$,

$t_1=1-\frac{1}{(2n+1)\pi}$

We can see that:

$\lim_{n\to\infty}|t_0-t_1|=\lim_{n\to\infty}|\frac{1}{(2n+1)\pi}-\frac{1}{2\pi n}|=\lim_{n\to\infty}|...|=0$

But:

$|f(t_0)-f(t_1)|=|\cos(t_0)-\cos(t_1)|=|\cos(2\pi n)-\cos((2n+1)\pi)|=2$

Therefor for every $0<\varepsilon<2$ $\nexists \delta$, so that for all $|t-t'|<\delta$ will be held: $|f(t)-f(t')|<\varepsilon$

b)

$D=\{(x,y)| 3 \leqslant x^2+y^2\leqslant 4\}$ is closed and bounded, the function $f(x,y)$ is continuous function in $D$, hence from Cantor's theorem we can deduce that $f(x,y)$ is uniformly continuous function.

2. It's correct. In a), I think you have to denote the new $f$ by $g$ because it's not the same function (in particular $f$ has two variables whereas $g$ has only one). You can denote $s_n$ instead of $t_0$ and $t_n$ instead of $t_1$ to show that those two numbers depend on $n$.