Answer
$${\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left[ {0,\infty } \right)$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left\langle {8,\sqrt t ,\root 3 \of t } \right\rangle \cr
& {\text{Let the vector function }}{\bf{r}}\left( t \right) = f\left( t \right){\bf{i}} + g\left( t \right){\bf{j}} + h\left( t \right){\bf{k}} \cr
& {\text{The component functions are:}} \cr
& f\left( t \right) = 8{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& g\left( t \right) = \sqrt t ,{\text{ Is continuous for }}t \geqslant 0,{\text{ }}\left[ {0,\infty } \right) \cr
& h\left( t \right) = \root 3 \of t \space {\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& {\text{Therefore,}} \cr
& {\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left[ {0,\infty } \right) \cr} $$
