Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ with $f(0,0)=0$ is a continuous function using epsilon-delta. [duplicate]

real-analysis multivariable-calculus continuity

Here is how you advance

$$ \frac{xy^2}{x^2+y^2} \leq \sqrt{x^2+y^2} < \epsilon=\delta.$$

Note: we used the inequality

$$ |a| \leq \sqrt{a^2+b^2}. $$

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