What does it mean by smooth curve in differential geometry?

Question

Suppose $\gamma :(a,b)\to \mathbb R^2$ be a parametrized curve with parameter $t\in (a,b)$($t$ can be compared with time).Now what does it mean when we say $\gamma$ is a smooth curve,does it mean that $\frac{d^n(\gamma)}{dt^n}$ exists for all $n\in \mathbb N$ i.e. $\gamma$ is infinitely differentiable with respect to $t$.I am a little confused because suppose the curve is $y=x|x|$ on $x\in \mathbb R$.Then $y$ is only one time differentiable w.r.t $x$ but does it guarantee that $\gamma(t)=(t,t|t|)$ is is one time differentiable w.r.t $t$?

Answer 1

A smoothness property for a function with values in $\mathbb R^n$, by definition, is equivalent to that same smoothness property holding simultaneously for each of its $n$ coordinate functions.

Since the two coordinate functions of $\gamma(t)$ are $x(t)=t$, which is smooth, and $y(t) = t \, |t|$, which is one time differentiable with respect to $t$, it follows that $x(t)$ and $y(t)$ are both one time differentiable with respect to $t$, and therefore by definition $\gamma(t)$ is one time differentiable with respect to $t$.