The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:

Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\epsilon}(M)$ for some $\epsilon>0$ (it means that $f,f^{-1}$ are differentiable, and the differential is $\epsilon$-H\"older continuous).

For all $l\in\mathbb{N}$ and $\chi>0$, define

$\Lambda_{\chi,l}:=\{x\in M:$ There is a splitting $T_xM=E^s(x)\oplus E^u(x)$, and $\forall v_u\in E^u(x),v_s\in E^s(x), n\geq0$:

$|d_xf^nv_s|\leq l e^{-\chi n}|v_s|$, $|d_xf^{-n}v_s|\geq l^{-1} e^{\chi n}|v_s|$ and

$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_u|\geq l^{-1} e^{\chi n}|v_u|$ $\}$

The following line in Newhouse's paper is that it is fairly easy to see that $E^u(x)$ change continuously for $x\in \Lambda_{\chi,l}$ on each piece with $\dim E^u(x)$ constant.

I've tried showing so myself, or finding some reference online- neither yielded results. I'm not sure I understand even what continuity in this context means (I'd rather not to go into Grassmannians or tangent bundles, and keep it analytic); therefore even a reduction to the continuity of $Jac(d_xf|_{E^u(x)})$ will be greatly appreciated. It is not clear to me if the continuity is unique to the sets $\Lambda_{\chi,l}$, or is it a property for all $\chi$-hyperbolic points, or all Lyapunov regular points, etc. As well as the regularity module- could one discuss H\"older continuity, uniform continuity etc. ?

Thanks ahead for anyone contributing... I'm sorry I couldn't present some more of my own progress on this question, I'm really quite lost on this one and would really like to understand his paper.