TITLE: Implicit function theorem with continuous dependence on parameter
QUESTION [10 upvotes]: Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$,  $f_p:= f|_{X\times \{p\}}:X\to Y$ is smooth.
Suppose also that $f_p\to f_{p_0}$ in $C^3_{loc}(X,Y)$ for $p\to p_0$ (i.e. once we fix an arbitrary compact $K\subset X$, we have uniform convergence in $C^3$-norm over it)
Suppose that $Df_{p_0}(x_0):X\to Y$ is an isomorphism, we would like to prove that under these assumptions there exists a continuous  implicit function, i.e.

Show that exists $U_{x_0}\times U_{p_0}\subset X\times P$ product neighbourhood of $(x_0,p_0)$ and continuous function $x:U_{p_0}\to U_{x_0}$ such that the zero locus of $f|_{U_{x_0}\times U_{p_0}}$ is the graph of $x$, $\{(x(p),p)\ | \ p\in U_{p_0}\}$.
If false please provide an example

Notice that the existence of the neighbourhood is already a difficult part. Indeed $C^1_{loc}$ convergence implies that there is $U_{p_0}$ such that for any $p\in U_{p_0}$, $Df_{p}(x_0)$ is isomorphism so we can apply the inverse function theorem but the size of the "inversion"-neighbourhood can shrink to zero as $p\to p_0$.
In order to prevent this, I thought to use the  $C^3_{loc}$ convergence, because we can estimate the radius of the neighbourhood using the second derivative in a neighbourhood of $x_0$, however the convergence is only on compact sets therefore I do not know how to use it to produce a bound for $D^2f_p(x)$ for $\|x-x_0\|<r$.
For the continuity I hope that somebody knows a version of the inverse function theorem or contraction principle that works in this non-smooth setting, unfortunately $C^r_{loc}(X,Y)$ is not even a metric space for $X,Y$ infinite dimensional.
$^1$ I am interested in the case where $P$ is a finite dimensional manifold so feel free to assume so if you can say something about this situation.

REPLY [2 votes]: While Pietro Majer has already provided an argument and answered the questions; here is a Paper reference for a more general parameter dependent implicit function Theorem (also has Differential dependence on parameters)
https://arxiv.org/abs/math/0303320
Glöckners Paper was published in the Israel Journal of math some years later.
Addendum (from my comment below): For the continuous parameterdepence take a look at Proposition 2.1. There the parameterspace $P$ is a topological space. If I remember correctly the main thrust of the theorem was to establish various differnetiability conditions of the implicit function depending on a parameter in a very general setting. SO this is the case discussed in the introduction. However, the basis is alsways the continuous result (but thanks for catching this Warlock of Firetop Mountain).