BCS Theory Energy Eigenstates By making the approimation, introduced by Cooper, that $V_{kk'}=-V$ for $k$ states out to a cutoff energy $\hbar \omega_c$ away from $E_F$ and that $V_{kk'}=0$ beyond $\hbar \omega_c,$ we can solve the Schrodinger equation for a two particle (singlet) wavefunction:
$$(E-2\epsilon_{\mathbf{k}})g_{\mathbf{k}}=\sum_{k'>k_f}V_{\mathbf{kk'}}g_{\mathbf{k'}}$$
becomes
$$g_{\mathbf{k}}=V\frac{\sum g_{k'}}{2 \epsilon_{k}-E}$$
[Michael Tinkham, Introduction to Superconductivity, Page 44-45.]
The author then replaces the sum by an integral,
$$\frac{1}{V} = \sum_{k>k_F}(2 \epsilon_{\mathbf{k}}-E)^{-1}\approx N(0)\int_{E_F}^{E_F + \hbar\omega_c} \frac{d \epsilon}{2\epsilon -E}$$
In general, a sum can be estimated by an integral
$$\sum_k f(k) \approx N\cdot\int_k dk\, f(k) $$
where $N$ denotes the number of times the variable $k$ is indexed in the sum per unit interval. In other words, if $k=[4,5,6,7,...]$ then $N=1,$ but if $k=[3.3,3.4,3.5,...]$ then $N=10.$ The author here though has changed the integration measure to $d\epsilon$. I take this to imply that it is the $\epsilon$ which are "evenly spaced" in the sum, rather than the $k.$ However, how can this be shown at this point?
 A: The correct mapping would be the following:
$$
\sum_k \dots \to \int d\varepsilon N(\varepsilon) \dots
$$
where $N(\varepsilon)$ is the density of states.
So the assumption that the author makes is to consider $N(\varepsilon) \approx N(0)$ for all $\varepsilon$ in the integration interval.
This is fine as long as $\hbar \omega_c \ll E_F$ and as long as the electrons are free to move in space. If there is an underlying lattice of positive ions this is still ok for many cases, even though in some particular cases this might be false (see e.g. Van Hove singularities).
EDIT:
Why this mapping?
The density of states of an electronic system with dispersion relation $\varepsilon_k$ is defined as
$N(\varepsilon) = \sum_k \delta(\varepsilon - \varepsilon_k)$, where $\delta$ is the Dirac delta function.
The idea is that the graphic of the function $N(\varepsilon)$ is a histogram of the list of energy eigenvalues.
If now you are summing over $k$ a function that depends on $k$ through $\varepsilon_k$ [let's call it $F(\varepsilon_k)$], then
$$
\sum_k F(\varepsilon_k) = \sum_k \int d\varepsilon F(\varepsilon) \delta(\varepsilon - \varepsilon_k)
= \int d\varepsilon F(\varepsilon) \sum_k \delta(\varepsilon - \varepsilon_k)
= \int d\varepsilon F(\varepsilon) N(\varepsilon) 
$$
