A curve $\mathbf{r}(t)$ is considered to be smooth if its derivative, $\mathbf{r}'(t)$, is continuous and nonzero for all values of $t$. I have some concerns regarding the second condition:
Let's examine the curves $\mathbf{r}_1(t) = \langle t,t\rangle$ and $\mathbf{r}_2(t) = \langle t^3,t^3\rangle$. The former has a constant velocity, $\mathbf{r}_1'(t) = \langle 1,1\rangle$, while the latter has a varying velocity, $\langle 3t^2,3t^3 \rangle$, with $\textbf{r}_2(0) = \textbf{0}$. According to the definition, the curve traced by $\mathbf{r}_1$ is smooth, while the curve traced by $\mathbf{r}_2$ is not. However, both functions trace the same curve.
What might be causing my confusion here?