Answer
$${\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left( {0,\infty } \right)$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = 2{e^{ - t}}{\bf{i}} + {e^{ - t}}{\bf{j}} + \ln \left( {t - 1} \right){\bf{k}} \cr
& {\text{Let the vector function be }}{\bf{r}}\left( t \right) = f\left( t \right){\bf{i}} + g\left( t \right){\bf{j}} + h\left( t \right){\bf{k}} \cr
& {\text{The component functions are:}} \cr
& f\left( t \right) = 2{e^{ - t}},{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& g\left( t \right) = {e^{ - t}},{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& h\left( t \right) = \ln \left( {t - 1} \right),{\text{ Is continuous for }}t - 1 > 0 \to \left( {0,\infty } \right) \cr
& {\text{Therefore,}} \cr
& {\bf{r}}\left( t \right){\text{ is continuous on the interval }}\left( {0,\infty } \right) \cr} $$
