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
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Expert's answer
2022-04-22T02:49:02-0400
Since it is not clear how does the function look like, consider two cases:
"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".
"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)."
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