Give an example of the following with justification 
i) A vector-valued function  which is not differentiable at .
ii) A function which is Legesgue measurable on R.

i) An example of a vector-valued function \( f : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) that is not differentiable at \((0,0,0)\) is the function defined by:
\[ f(x, y, z) = \begin{cases} 
      (0, 0, 0) & \text{if } (x, y, z) = (0, 0, 0) \\
      \left(\frac{xyz}{x^2+y^2+z^2}, \frac{xyz}{x^2+y^2+z^2}, \frac{xyz}{x^2+y^2+z^2}\right) & \text{if } (x, y, z) ___ ___ ________ ____ ___ _____ ____ _____ ___ _________.
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