The relation between continuity and differentiability of binary function For example, if f (x, y) is continuous at point (0,0), then if x and y are all close to 0, f (XY) / (| x | y |) exists, is f (XY) differentiable at point (0,0) I know how to smoke The denominator is (| x | plus | y |)

The relation between continuity and differentiability of binary function For example, if f (x, y) is continuous at point (0,0), then if x and y are all close to 0, f (XY) / (| x | y |) exists, is f (XY) differentiable at point (0,0) I know how to smoke The denominator is (| x | plus | y |)

The existence of 1 / 2 {[f (0 + X, + y) - f (0,0)] / | x | + [f (0 + X, + y) - f (0,0)] / | y |]} can be obtained from the known condition, that is, the existence of partial derivative of F (x, y) at the right side of point (0,0). The sufficient condition for differentiability is that the partial derivative of F (x, y) is continuous at point (x, y). The known condition only proves that the partial derivative is right continuous, not left continuous, so it is not differentiable