PX262 - C4c - continuous eigenvalues

previous arguments assume that the eigenvalues of the operators have discrete values
this is true for energy, but not for position and momentum
taking approximation by discrete steps, but also the limit $Δ \to 0$ , the eigenvalue equation becomes: $\hat{Q} ϕ (k, x) = q (k) ϕ (k, x)$ where, $k$ is a continuous variable that labels individual steps (like the subscript $n$ )
the equivalent wavefunction is:

ψ (x) = \int a (k) ϕ (k, n) d k

the probability of getting a result inside the range $d k$ in the vicinity of $k$ is:

p = | a (k) |^{2} d k

\int ϕ^{*} (k, x) ϕ (k^{'}, x) d x = δ (k - k^{'})

the general wavefunction can be written as a liner superposition of eigenfunctions as: $ψ (x) = \int a (k) ϕ (k, x) d k$