Stability of a differentiator
For a linear system, stability is usually defined in terms of the "bounded input/bounded output" (BIBO) criteria. For a pure integrator, the system is not stable, since any constant input (except for 0) creates an unbounded output.

Another way to look at it, is that a unit impulse response should have a bounded L1-norm. The impulse response of an integrator is the unit Heaviside function (0 when t &lt; 0, and 1 when t >= 0):
$$
r(t) = \int_{-\infty}^t\delta_0(\tau)d\tau= H(t) = \begin{cases}0,\qquad t &lt; 0\\ 1, \qquad t \ge 0\end{cases}
$$
But then 
$$
\| r \|_1 = \int_{-\infty}^\infty |r(t)| dt = \int_{-\infty}^\infty H(t)dt = \int_0^\infty dt = \lim_{T \to \infty}\int_0^T dt = \lim_{T\to\infty} T = \infty,
$$
So the L1-norm is not bounded.