Question

Determine the set of points at which the function is continuous.
1) F(x,y)=sin(xy)÷(e^x-y^2)
2) F(x,y)=x-y÷(1+x^2+y^2)
3) F(x,y)=arctan(x+√y)
4) F(x,y)=e^x^2y+√x+y^2
5) G(x,y)=ln(x^2+y^2-4)

Answer 1

Given first function

(i) $f(x,y)=\dfrac{sin(xy)}{e^x-y^2}$

$f(x,y)$ is continous for all values except at $e^x-y^2=0$

$\implies e^x=y^2$

Taking log on both sides

$\implies xlog e=2logy \\\implies x=2logy$

(ii) $f(x,y)=\dfrac{x-y}{1+x^2+y^2}$

$since$ $1+x^2+y^2>0$

$f(x,y)$ is continous of alll value of $x$ and $y$

(iii) $f(x,y)=arctan(x+\sqrt{y})$

$f(x,y)$ is continuous for all value of $x$ and $y$ except $y<0$

(iv) $f(x,y)=e^{x^2y}+\sqrt{x}+y^2$

Given function is continuous for all values of $x$ and $y$ except at $x<0$

(v) $f(x,y)=ln(x^2+y^2-4)$

Given function is continous for $x^2+y^2>4$

$\implies x\ge 0, y\ge 2$

So the set of values for which given function are cintinuous

are $x\ge 0,y\ge 0$