Let $H$ be a bounded linear operator on some separable Hilbert space such that $H$ has pure point spectrum on the interval $[a,b]$: $\sigma_{pp}(H)=[a,b]$. Note that here we define the pure point part of the spectrum as the closure of the set of eigenvalues. That means that a given point $x\in[a,b]$ may be either an eigenvalue or one could find an eigenvalue arbitrarily close to it.
So let $x\in[a,b]$ not be an eigenvalue of $H$. Is $\chi_{\{x\}}(H)=0$? If not, what does it project onto?
It has to be zero.
Note that for something like $\chi_{\{x\}}(H)$ to make sense, you need $H$ to be selfadjoint, or at least normal.
Since $t\,\chi_{\{x\}}(t)=x\,\chi_{\{x\}}(t)$, we have $H\chi_{\{x\}}(H)=x\,\chi_{\{x\}}(H)$. So if $\chi_{\{x\}}(H)y=y$ with $y\ne0$, we have $$ Hy=H\chi_{\{x\}}(H)y=x\,\chi_{\{x\}}(H)y=x\,y, $$ and $x$ is an eigenvalue.