Answer
$$\eqalign{
& \left( {\text{a}} \right){\text{ Domain: }}\left( {0,\infty } \right) \cr
& \left( {\text{b}} \right){\text{ Continuous for }}t > 0 \cr} $$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = \ln t{\bf{i}} + t{\bf{j}} + t{\bf{k}} \cr
& {\text{Let the vector function be }}{\bf{r}}\left( t \right) = f\left( t \right){\bf{i}} + g\left( t \right){\bf{j}} + h\left( t \right){\bf{k}} \cr
& {\text{The component functions are:}} \cr
& f\left( t \right) = \ln t,{\text{ Is continuous for }}t > 0 \cr
& g\left( t \right) = t,{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& h\left( t \right) = t,{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr
& {\text{Intersecting the domains we obtain: }}\left( {0,\infty } \right) \cr
& \left( {\text{a}} \right){\text{ Domain: }}\left( {0,\infty } \right) \cr
& \left( {\text{b}} \right){\text{ Continuous for }}t > 0 \cr} $$
