Question

Check whether the function `f(x,y)={(6xy)/sqrt(x^2+y^2) " if " (x,y)≠(0,0), {0 if (x,y)=(0,0)` is continuous at the origin?

Answer 1

See below.

Explanation:

If `abs(f(x_1)-f(x_2)) le k abs(x_1-x_2)` for all `x_2` then `f(x)` is said continuous at `x_1`

Now

`abs(f(0,0)-f(x,y)) le abs(0-(6xy)/sqrt(x^2+y^2)) le 6 abs(0-x)`

or

`abs((6xy)/sqrt(x^2+y^2)) le 6 abs x` and also

`abs((6xy)/sqrt(x^2+y^2)) le 6 abs y`

and then

`sqrt2abs((6xy)/sqrt(x^2+y^2))le 6 sqrt(x^2+y^2)`

hence

`f(x,y)` is continuous at `(0,0)`