## Thomas' Calculus 13th Edition

$f(t), g(t), h(t)$ is continuous at $t=t_0$ so, $r(t)$ is also continuous at $t=t_0$
Consdier $r(t)=\lt f(t), g(t), h(t) \gt$ Then, we have $r'(t)=\lt f'(t), g'(t), h'(t) \gt$ Here, $r'(t)$ is continuous and differentiable at $t=t_0$ this means that $f(t), g(t), h(t)$ also differentiable at $t=t_0$ we can conclude that $f(t), g(t), h(t)$ is continuous at $t=t_0$ so, $r(t)$ is also continuous at $t=t_0$