Given that f(x, y)...

Question

Given that f(x, y) = { (xy) / (x² + y²) if (x, y) ≠ (0, 0)

0 if (x, y) = (0, 0)

Is f(x, y) continuous at (0, 0)? Explain.

Answer 1

It is not continuous at (0,0) because the lim(f(x,y)) as x,y-> 0 does not even exist.

Pick any constant k and approach (0,0) along the curve y = kx.

On that curve

f(x,y) = kx²/(x² + (kx)²) = k/(1 + k²)

For each different values of k the limit is different.