You need to look at the potential in momentum space. The Fourier transform gives: \begin{align} \hat V(k) &:= \int dx V(x) e^{-ikx} \\ &= 2\pi\left(V_0\delta(k)+V_1\left(\delta\left(k-\frac{2\pi}{a}\right)+\delta\left(k+\frac{2\pi}{a}\right)\right)+...\right) \end{align} By analogy with the discrete case, these Dirac deltas (apart from the first one) give "off diagonal" coefficients in Fourier space. In particular, starting from the usual Hamiltonian: $$ H_0 = \frac{k^2}{2m} $$ the states $|k=\pm\pi/a\rangle$ are degenerate eigenvectors with $H_0$, but get mixed due to $V$, lifting the degeneracy. This will generate an energy splitting.

Note that without further calculation, you cannot know whether this energy splitting at the edge of the Brillouin zone corresponds to a separation of energy bands, or rather an overlap of two energy bands (how to connect the splitter energy levels with the non split energy levels for $|k|<\pi/a$ and $|k|>\pi/a$).

Hope this helps.

You need to look at the potential in momentum space. The Fourier transform gives: \begin{align} \hat V(k) &:= \int dx V(x) e^{-ikx} \\ &= 2\pi\left(V_0\delta(k)+V_1\left(\delta\left(k-\frac{2\pi}{a}\right)+\delta\left(k+\frac{2\pi}{a}\right)\right)+...\right) \end{align} By analogy with the discrete case, the $V_1$ Dirac deltas give "off diagonal" coefficients in Fourier space. In particular, starting from the usual Hamiltonian: $$ H_0 = \frac{k^2}{2m} $$ the states $|k=\pm\pi/a\rangle$ are degenerate eigenvectors with $H_0$, but get mixed due to $V$, lifting the degeneracy. This will generate an energy splitting.

The splitting leads to the creation of bands. However, there are two possible cases, and without further calculation, it's hard to determine which one is relevant. In one case, this energy splitting at the edge of the Brillouin zone corresponds to a separation of energy bands (lower level of the split connects with the energies of $|k|<\pi/a$). In another case case, it corresponds to an overlap of energy bands (higher level of the split connects with the energies of $|k|<\pi/a$). It turns out that the former is correct, and can be checked by doing a second order perturbation calculation in $V$. Check out for example D. Tong's Lecture notes for the full computation.

Hope this helps.