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 problem with the function $f(t) = t^{1/3}$ is not that it is something else. The problem is that in the point $0$, the slope of the curve is infinite as the function turns completely vertical. The curve drawn is actually smooth, as you can find a smooth parametrisation of it.
The parametrisation $$t\rightarrow (t, \sqrt[3]{t})$$ is not smooth, however, you can reparametrise the curve as $$t\rightarrow (t^3, t)$$ which is obviously smooth.
