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).
"f(x,y)=\\begin{cases}\n \\frac{2xy}{x^2+y^2} &(x,y) \\neq (0,0) \\\\\n 0 &(x,y)=(0,0)\n\\end{cases}"
Choosing the path x = 0 we see that f (0, y) = 0, so
"\\displaystyle\\lim_{y\u21920} f (0, y) = 0"
Choosing the path x = y we see that
"f (x, x) = 2x^ 2 \/2x^ 2 = 1"
so
"\\displaystyle\\lim_{x\u21920} f (x,x) = 1"
The Two-Path Theorem (if a function has two different limits along two different paths) implies that
"\\displaystyle\\lim_{(x,y)\u2192(0,0)} f (x,y)"
does not exist.
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