Intuition behind smooth functions. Smooth functions $f(t)$ are those such that $\frac{d^nf(t)}{dt^n}$ exists for all $n\in\Bbb{N}$. 
I understand the intuition behind smoothness for functions like $f(t)=| t|$ and $f(t)=\sqrt{t}$. $f(t)$ has a "sharp" (and hence non-smooth) turn at $t=0$. Similarly, $f(t)=\sqrt{t}$ ends abruptly at $t=0$ (and hence is not "smooth"). 
However, functions like $f(t)=t^{\frac{1}{3}}$ seem "smooth" enough to me! 

Why does the intuitive understanding of smoothness fail here? Is this just another case of extending a definition of a term to non-intuitive cases?
Thanks in advance!

The curve itself admits a smooth parametrization, $\{(t^3,t):t\in\Bbb R\}.$ The function $f(t)=t^{1/3}$ on the other hand is not differentiable at $t=0$; one way of thinking about this is that the function $f(t)$ parametrizes the curve in such a way that we rush through the origin with infinite speed, and that isn't allowed on a smooth ride. Thus, the problem is with the parametrization, not the geometry.
