$f(x,y)=\arcsin \frac{x}{y}$ is continuous but not uniformly continuous in its domain

Question

I need to prove that $f(x,y)=\arcsin \frac{x}{y}$ is continuous, but not uniformly continuous on its domain. I noticed that the domain of the function is $D_f=\{(x,y)|-y\leq x \leq y$ if $y>0$, and $y\leq x \leq -y$ if $y<0\}.$ I started to prove the continuity by the epsilon-delta deffinition, but I'm stuck at proving that $|\arcsin \frac{x}{y} - \arcsin \frac{x'}{y'}|<\epsilon$.

Answer 1

To show that the function $f$ is not unifromly continous first recall that $\operatorname{arcsin} 1 =\tfrac{\pi}2$ and $\operatorname{arcsin} \tfrac 12 =\tfrac{\pi}6$. Thus for each natural $n$ we have $f\left(\tfrac 1n, \tfrac 1n\right)= \tfrac{\pi}2$ and $f\left(\tfrac 1n, \tfrac 2n\right)= \tfrac{\pi}6$, whereas $\left|\left(\tfrac 1n, \tfrac 1n\right)- \left(\tfrac 1n, \tfrac 2n\right)\right|=\tfrac 1n$.