Question

Is the function continuous? Explain your Answer:

f(x,y)=1 if (x,y)=(0,0) and f(x,y) = \[\frac{(x+y)^{2}}{\sqrt{x^2+y^2}}\] if (x,y) \[\neq\] (0,0)

Answer 1

sorry i'm not sure how to type out a piecewise function

Answer 2

\[\lim_{(x,y) \rightarrow (0,0)}(x+y)^2/\sqrt{x^2+y^2}\]
along y=x
\[\lim_{(x,x) \rightarrow (0,0)}4x^2/(x \sqrt{2)}=\lim_{(x,x) \rightarrow (0,0)}(2\sqrt{2})x=0 \neq f(0,0) =1\]
The limit from at least one direction does not equal the value at the point it is approaching, so the function is no continuous at (x,y)=(0,0)

I think...