What function can be differentiated twice, but not 3 times?
Solution 1:
What function cannot be differentiated $3$ times?
Take an integrable discontinuous function (such as the sign function), and integrate it three times. Its first integral is the absolute value function, which is continuous: as are all of its other integrals.
Solution 2:
$f(x) =\begin{cases} x^3, & \text{if $x\ge 0$} \\ -x^3, & \text{if $x \lt 0$} \\ \end{cases}$
The first and second derivatives equal $0$ when $x=0$, but the third derivative results in different slopes.