Question

Check whether the function f(x,y)={4x^2y/4x^4+y^2 ,(x,y)/=(0,0)
{0,(x,y)=0 is continuous at(0,0)

Answer 1

Consider the sequence $(x_k, y_k) = (\frac 1k,\frac 1{k^2})$ : $f(x_k, y_k)=\frac{\frac 4{k^4}}{\frac 5{k^4}} =\frac 45 \rightarrow \frac 45 \not =0=f(0,0), k \to \infin.$

Consequently, the function $f(x,y)$ is not continuous at $(0,0)$ .