Answer
$${\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left( { - \frac{\pi }{2} + n\pi ,\frac{\pi }{2} + n\pi } \right)$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \left\langle {{e^{ - t}},{t^2},\tan t} \right\rangle \cr
& {\text{Let the vector function }}{\bf{r}}\left( t \right) = f\left( t \right){\bf{i}} + g\left( t \right){\bf{j}} + h\left( t \right){\bf{k}} \cr
& {\text{The component functions are:}} \cr
& f\left( t \right) = {e^{ - t}},{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& g\left( t \right) = {t^2},{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& h\left( t \right) = \tan t,{\text{ Is not continuous for }}\left\{ {\frac{\pi }{2} + n\pi } \right\},{\text{ }}n{\text{ integer}} \cr
& {\text{Therefore,}} \cr
& {\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left( { - \frac{\pi }{2} + n\pi ,\frac{\pi }{2} + n\pi } \right) \cr} $$
