Consider a $L^2$ map $f:\Omega \to \mathbb{R}$ for some compact $\Omega \subset \mathbb{R}^d$. Using the $k$-means clustering algorithm, we decompose $\Omega$ into $k$ regions $\Omega_1,\dots,\Omega_k$ for which $f$ is constant within each region. Let $f_j \in \mathbb{R}^s$ be that constant value for each $\Omega_j$. The "clustered" function can be just written as a piecewise uniform function: \begin{equation} f^c(x) = \sum_{i=1}^k\chi_{\Omega_i}(x)f_i \end{equation} with $\chi$ being the indicator fucntion. I observe that $f^c$ is also $L^2$. I was just wondering what we can say about the operator $\Phi: L^2(\Omega) \to L^2(\Omega)$ given by $f \mapsto f^c$. Is it continuous? Injective?
Also, what about if we let $f$ not be $L^2$ but in the Sobolev space $H^s$? What can we say about the operator $\Phi: H^2(\Omega) \to L^2(\Omega)$ defined in the identical way?