Continuity of a piecewise function

#piecewise#continuity#xy

Statement

Show that $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ (with $f(0,0)=0$ ) is continuous at $(0,0)$ .

Hint

$|xy|\le\frac12(x^2+y^2)$ .

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Bound. $|f(x,y)| = \frac{|xy|}{\sqrt{x^2+y^2}} \le\frac12\sqrt{x^2+y^2}\to0$ .
Conclusion. $f$ is continuous at $(0,0)$ .