Consider the class of $C^0$ signals $x(t)$ such that
$$
\left|x\left(t+T\right)-x\left(t\right)\right|\leq\epsilon 
$$
I would like to find a uniform bound on the derivative of its moving rms, which is defined by
$$
J\left(t\right)=\left(\frac{1}{T}\int\_{t-T}^{t}\left|x\left(\tau\right)\right|^{p}d\tau\right)^{\frac{1}{p}}
$$
with $p=2$. Apparently, when $\epsilon=0$, $J$ is always constant and the derivative is uniformly 0; so I wonder whether $
\left|\frac{dJ}{dt}\right|
$ is uniformly small when $\epsilon$ is small.

It might be worth noting that a uniform bound could be found when $p=1$ (in the case of moving average), see <https://math.stackexchange.com/q/57307>.

It would also be interesting to find out whether it holds for other $p\in[1,\infty]$.