Consider a line $L(\theta)=t(cos(\theta), sin(\theta)), t\in \mathbb{R}$, then $v(\theta)=\left( \begin{array}{cc} cos(\theta) \\ sin(\theta)\end{array}\right)$ is an extrinsic basis of the line in $\mathbb{R}^2$ and the correspondence $\theta \to v(\theta)$ is discontinuous.
While taking an intrinsic spanning set $B(\theta)=\left\lbrace cos(\theta)\left( \begin{array}{cc} cos(\theta) \\ sin(\theta)\end{array}\right) , sin(\theta)\left( \begin{array}{cc} cos(\theta) \\ sin(\theta)\end{array}\right)\right\rbrace$ for canonical co-ordinate system of the line, then the correspondence $\theta \to B(\theta)$ is continuous.