Answer to Question #275098 in Calculus for Parth

Question

Give an example of a function of two variables such that f(0,0) = 0 but f is NOT continuous at (0,0). Explain why the function f is NOT continuous at (0,0).

Answer 1

Consider the function $f:\R\times\R\to\R,\ f(x,y)=\begin{cases}0, \text { if }(x,y)=(0,0)\\ 1, \text { if }(x,y)\ne(0,0)\end{cases}.$

Taking into account that $\lim\limits_{x\to 0,\\ y\to 0}f(x,y)=1\ne 0=f(0,0),$ we conclude that the function is not continuous at $(0,0).$