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 $\epsilon$-Hölder 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
$|d_xf^{-n}v_u|\leq l e^{-\chi n}|v_u|$, $|d_xf^{n}v_s|\geq l^{-1} e^{\chi n}|v_u|$ $\}$
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 Hölder 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.
It is unavoidable to go to Grassmannians to make it rigorous, but you can disguise it by using only bases. Namely, if a sequence of bases converges to a base, then you also have continuity in the Grassmannian. So (almost) everybody uses only convergence of bases (let's hope that they know how to complete the argument).
You may have a look at the book Dynamical Systems by Barreira and Valls for an argument using the distance on the Grassmannian. The argument is somewhat involved, although personally I would agree with Sheldon. Note that you actually conclude a posteriori that the stable and unstable dimensions are locally constant (I don't know how to show that a priori, but maybe there is some argument, although we don't need for what you ask).
PS. Note that also the hypothesis of $f$ having Hölder derivative is not needed there. You only need the four inequalities together with the finite-dimensionality of the manifold.