Perplexing (for me at least) statement from the site: http://www.quora.com/Mathematics/What-are-some-of-the-most-counterintuitive-mathematical-results
"Fact: You can have a car stand still for hours and then start moving at noon, but its speed, acceleration, meta-acceleration (the rate of change of acceleration), meta-meta-acceleration and so forth were all 0 at noon. How did it then start moving? Miracle."
How is this possible? Is it suggesting/joking with "Miracle", that the velocity function is non-continuous at noon (step function maybe?), or is there some strange solution to
$$ds/dt = 0$$
which isn't a constant $s = c$
Note (I don't know if this image belongs specifically to the above "fact", but it's put right below it on the website):
http://qph.is.quoracdn.net/main-qimg-630e434b5a338247e62b576f73034e24?convert_to_webp=true
The position as a function of time can be something like
$$ s(t) = \begin{cases} 0 & t\le 0 \\ e^{-1/t} & t > 0 \end{cases} $$
This is $\mathcal C^\infty(\mathbb R)$, and all derivatives at $t=0$ are $0$.
To prove this, note that every higher derivative in the $t>0$ region has the form $e^{-1/t}P(1/t)$ for some polynomial $P$. Once that is dealt with, it is not too difficult to see that every higher deriviative exists (and is $0$) at $t=0$ too.