As far as I knew and as stated in this post smoothness of multivariable functions
A function $f:\mathbb{R}^n \to \mathbb{R}^m$ is smooth if and only if all of its components are smooth.
Consider a curve $\gamma(t)=(t^2,t^3)$ . This is just a multivariable function for n=1, m=2. But it is a well-know example of a curve that is not smooth, because $y=t^3, x=t^2$ yields $x=y^{2/3}$ is not smooth at the origin. $\gamma(t)$ and $x(y)$ are the same function written in different ways, aren't they?
On the other hand the components of $\gamma$ are $t^2$ and $t^3$ which are smooth so how isn't this a contradiction with the definition of smoothness above?