Discontinuous function by epsilon delta How do I prove that the function $ \frac{x ^ 2 y} {| x ^ 3 | + y ^ 2} $ is not continuous on $ \{0,0\} $ using epsilon-delta?
I have tried it using sequences that have $ \{0,0 \} $, but I would like an idea to test it by epsilon-delta, since it seems that the solution is complicated by that means

$ \frac{x ^ 2 y} {| x ^ 3 | + y ^ 2} $  is what you typed in.
Note $$  0 \leq \left( |y| -|x|^{\frac{3}{2}}  \right)^2 = y^2 - 2 |y| |x|^{\frac{3}{2}}  + |x|^3  $$
so
$$    2 |y| |x|^{\frac{3}{2}} \leq y^2 + |x|^3 $$
and
$$ \frac{ 2 |y| |x|^{\frac{3}{2}}  }{ |x|^3 + y^2 }   \leq 1   $$
as long as $x,y$ are not both zero.
Thus
$$ \frac{ 2 |y| x^2  }{ |x|^3 + y^2 }   \leq \sqrt{|x|}   $$
This means that assigning the value $0$ as the value of the function at $(0,0)$ creates a continuous function.
