Question

state the statement is true or false the function f[x,y]={x^2y/x^4+y^2[x,y]=0 is not continuous at [0,0] and 0,[x,y]=0

Answer 1

Since it is not clear how does the function look like, consider two cases:

1. $f(x,y)=\frac{x^2y}{x^4}+y^2$, $f(0,0)=0$. Consider the line $y=x$. We receive $f(x,x)=\frac1x+x^2$. The function tends to $\infty$ as $(x,y)$ approaches $(0,0)$ along $y=x.$ Thus, the function is not continuous at $(0,0)$ and it is not possible to define it in such a way that it will be continuous at $(0,0)$, since it approaches $\infty$ as $x$ approaches along the line $y=x$.
2. $f(x,y)=\frac{x^2y}{x^4+y^2}$, $f(0,0)=0.$ Consider the parabola $y=x^2$. We receive $f(x,x^2)=\frac12$. On the other hand, consider the line $y=x$. We receive: $f(x,x)=\frac{x^3}{x^4+x^2}$. The latter approaches as $x$ approaches . Thus, the function is not continuous at $(0,0)$ and it is not possible to define it in such a way that it will be continuous, because it tends to different values along different curves as $(x,y)$ approaches $(0,0).$

Thus, the statement is true in both cases.