Consider a
$C^{2}$
family of mixing
$C^{4}$
piecewise expanding unimodal maps
$t\in [a,b]\mapsto f_{t}$
, with a critical point
$c$
, that is transversal to the topological classes of such maps. Given a Lipchitz observable
$\unicode[STIX]{x1D719}$
consider the function
$$\begin{eqnarray}{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)=\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t},\end{eqnarray}$$
where
$\unicode[STIX]{x1D707}_{t}$
is the unique absolutely continuous invariant probability of
$f_{t}$
. Suppose that
$\unicode[STIX]{x1D70E}_{t}>0$
for every
$t\in [a,b]$
, where
$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{t}^{2}=\unicode[STIX]{x1D70E}_{t}^{2}(\unicode[STIX]{x1D719})=\lim _{n\rightarrow \infty }\int \left(\frac{\mathop{\sum }_{j=0}^{n-1}\left(\unicode[STIX]{x1D719}\circ f_{t}^{j}-\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t}\right)}{\sqrt{n}}\right)^{2}\,d\unicode[STIX]{x1D707}_{t}.\end{eqnarray}$$
We show that
$$\begin{eqnarray}m\left\{t\in [a,b]:t+h\in [a,b]\text{ and }\frac{1}{\unicode[STIX]{x1D6F9}(t)\sqrt{-\log |h|}}\left(\frac{{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t+h)-{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)}{h}\right)\leqslant y\right\}\end{eqnarray}$$
converges to
$$\begin{eqnarray}\frac{1}{\sqrt{2\unicode[STIX]{x1D70B}}}\int _{-\infty }^{y}e^{-\frac{s^{2}}{2}}\,ds,\end{eqnarray}$$
where
$\unicode[STIX]{x1D6F9}(t)$
is a dynamically defined function and
$m$
is the Lebesgue measure on
$[a,b]$
, normalized in such way that
$m([a,b])=1$
. As a consequence, we show that
${\mathcal{R}}_{\unicode[STIX]{x1D719}}$
is not a Lipchitz function on any subset of
$[a,b]$
with positive Lebesgue measure.