Question: Are the following functions \({{\mathbb{R}}^{3}}\to \mathbb{R}\) continuous at \(\left( x,y,z \right)=\left( 0,0,0 \right)\) ?

(a)

\[{{f}_{1}}\left( x,y,z \right)=\left\{ \begin{aligned} & \frac{x}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\,\,\,\,\,\,}\,\,\,\,\left( x,y,z \right)\ne \left( 0,0,0 \right) \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( x,y,z \right)=\left( 0,0,0 \right) \\ \end{aligned} \right.\]

(b)

\[{{f}_{2}}\left( x,y,z \right)=\left\{ \begin{aligned} & \frac{xyz}{\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\,\,\,\,\,\,}\,\,\,\,\left( x,y,z \right)\ne \left( 0,0,0 \right) \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( x,y,z \right)=\left( 0,0,0 \right) \\ \end{aligned} \right.\]