I am trying to understand what it means for a curve to be smooth. Intuitively, it would seem that a curve with a sharp bend is not smooth, for example: $t \mapsto \left[\begin{array}{c}t \\ |t| \end{array}\right] ~~ t \in [-1,1]$ But, this can be reparametrized to be smooth: $t \mapsto \left[\begin{array}{c}t^3 \\ -t^3 \end{array}\right]~~t \in[-1,0];~~~ t \mapsto \left[\begin{array}{c}t^3 \\ t^3 \end{array}\right]~~t \in[0,1] $
Is this true in general or is this an instance of confusing the image of a curve with the curve.