# Few functions and I have to study the following aspects: Continuity in the point (0,0), If the derivative exists at (0,0), Continuity of the partial derivatives at (0,0), Directional derivatives at (0,0)

Few functions and I have to study the following aspects:
Continuity in the point (0,0)
If the derivative exists at (0,0)
Continuity of the partial derivatives at (0,0)
Directional derivatives at (0,0)
One of the functions is, for example:
$f\left(x,y\right)=\left\{\begin{array}{ll}\frac{{x}^{2}{y}^{2}}{\sqrt{\left(}{x}^{2}+{y}^{2}\right)},& \text{if if (x,y) not (0,0)}\\ 0,& \text{if (x,y) = (0,0)}\end{array}$
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Use the polar coordinates transformation:
$\left\{\begin{array}{lll}x& =& r\mathrm{cos}\theta \\ y& =& r\mathrm{sin}\theta \end{array}$
Then, $f\left(x,y\right)={r}^{3}{\mathrm{cos}}^{2}\theta {\mathrm{sin}}^{2}\theta$
Now, since ${\mathrm{cos}}^{2}\theta {\mathrm{sin}}^{2}\theta$ is bounded, one can use the squeeze theorem.