Given an analytic manifold $M\subset \mathbb{R}^n$, analytically diffeomorphic to the closed hypercube $[-1,1]^{dim(M)}$, for $x\in\mathbb{R}^n$ close enough to $M$ there is a unique $x_0=\text{arg}\min_{y\in M}d(x,y)$. If $x_0$ is an interior point of $M$ and we change $x$ analytically does $x_0$ change analytically also? More formally, if $x(t)$ is a real analytic path in $\mathbb{R}^n$ with $x(0)=x$ is the corresponding $x_0(t)$ analytic in $\mathbb{R}^n$ in a neighhbourhood of $t=0$? Is there a situation where $x_0$ changes analytically in the manifold but not if we view the manifold as embedded in $\mathbb{R}^n$?
My work so far has been based around the Implicit Function Theorem, but it hasn't gotten me anywhere.