Chapter 12 - Vector-Valued Functions - Review Exercises - Page 863: 2

$$\eqalign{ & \left( {\text{a}} \right){\text{ Domain: }}\left[ {0,4} \right) \cup \left( {4,\infty } \right) \cr & \left( {\text{b}} \right){\text{ Continuous for }}\left[ {0,4} \right) \cup \left( {4,\infty } \right) \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = \sqrt t {\bf{i}} + \frac{1}{{t - 4}}{\bf{j}} + {\bf{k}} \cr & {\text{Let the vector function be }}{\bf{r}}\left( t \right) = f\left( t \right){\bf{i}} + g\left( t \right){\bf{j}} + h\left( t \right){\bf{k}} \cr & {\text{The component functions are:}} \cr & f\left( t \right) = \sqrt t ,{\text{ Is continuous for }}t \geqslant 0 \cr & g\left( t \right) = \frac{1}{{t - 4}},{\text{ Is continuous for all real numbers except }}t = 4 \cr & h\left( t \right) = 1,{\text{ Is continuous for all real numbers: }}\left( { - \infty ,\infty } \right) \cr & {\text{Intersecting the domains we obtain:}} \cr & \left[ {0,4} \right) \cup \left( {4,\infty } \right) \cr & \left( {\text{a}} \right){\text{ Domain: }}\left[ {0,4} \right) \cup \left( {4,\infty } \right) \cr & \left( {\text{b}} \right){\text{ Continuous for }}\left[ {0,4} \right) \cup \left( {4,\infty } \right) \cr} $$