mathcal{F}$ that satisfies this equation for all possible $\mu\in V(\mu)$ and $\nu\in\mathbb{R}-V(\mu)$ is $\mathcal{F}_{\nu}(\mu)=0$ which means, according to Table \[Table: spectrum\], that the point spectrum of $\mu$ is empty and $(\mu-\nu)^{-1}$ exists for all $\nu$. When $\nu\in V(\mu)$, however, this inverse is not continuous and we show below that in $\textrm{Map}(V(\mu),0)$, $\nu\in V(\mu)$ belongs to the continuous spectrum of $\mu$. This distinction between the nature of the inverses depending on the relative values of $\mu$ and $\nu$ suggests a wider “non-function” space in which to look for the solutions of operator equations, and in keeping with the philosophy embodied in Fig. \[Fig: GenInv\] of treating inverse problems in the space of multifunctions, we consider all $\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R}))$ satisfying Eq. (\[Eqn: eigen\]) to be eigenfunctions of $\mu$ for the corresponding eigenvalue $\nu$, leading to the following multifunctional solution of (\[Eqn: eigen\])$$\begin{aligned}
\mathscr F_{\nu}(\mu) & = & \left\{ \begin{array}{ccl}
(V(\mu),0), &  & \textrm{if }\nu\notin V(\mu)\\
(V(\mu)-\nu,0)\bigcup(\nu,\mathbb{R})), &  & \textrm{if }\nu\in V(\mu),\end{array}\right.\end{aligned}$$

where $V(\mu)-\nu$ is used as a shorthand for the interval $V(\mu)$ with $\nu$ deleted. Rewriting the eigenvalue equation (\[Eqn: eigen\]) as $\mu_{\nu}(\mathscr F_{\nu}(\mu))=0$ and comparing this with Fig. \[Fig: GenInv\], allows us to draw the correspondences $$\begin{aligned}
f & \Longleftrightarrow & \mu_{\nu}\nonumber \\
X\textrm{ and }Y & \Longleftrightarrow & \{\mathscr F_{\nu}\in\textrm{Multi}(V(\mu),\mathbb{R})\!:\mathscr F_{\nu}\in\mathcal{D}(\mu_{\nu})\}\nonumber \\
f(X) & \Longleftrightarrow & \{0\!:0\in Y\}\label{Eqn: GenInv_Spectrum}\\
X_{\textrm{B}} & \Longleftrightarrow & \{0\!:0\in X\}\nonumber \\
f^{-} & \Longleftrightarrow & \mu_{\nu}^{-}.\nonumber \end{aligned}$$ Thus a multifunction in $X$ is equivalent to $0$ in $X_{\textrm{B}}$ under the linear map $\mu_{\nu}$, an