The notion that flux needs to be continuous is false. Flux can be rightly viewed as the amount field lines emanating from a certain region. And the number of those field lines is proportional to the charge enclosed in that region. This, mathematically stated, is the Gauss law:
$$ \int_{\mathcal S} \mathbf E\cdot \mathrm d \mathbf s= \frac{q_{\rm enclosed}}{\varepsilon_0}$$
When we apply the above formulation of Gauss law to an infinitesimal region (volume element), we get the first Maxwell equation:
$$\nabla\cdot\mathbf E=\frac{\rho}{\varepsilon_0}$$
Viewing flux as the "flow" of field lines is completely fallacious, since there is nothing that's flowing in reality. The electrostatic field is static. Nothing is flowing at all. The arrows on the field lines only indicate the direction of the field, not the flow of anything. Viewing flux as the amount of field lines emerging/emanating from a region is paradox-free, and true in all cases.
In the example given in the figure, there are some induced charges on the surface of the consuctor, thus there must be net flux emanating or entering any region containing those charges (of course, the region needs to contain a net charge, so you cannot choose a region which encloses zero net charge). In this case, taking a long thin strip parallel to the conductors surface, enclosing one of the edges shows that, indeed, there is a net flux entering or exiting that surface, as expected.
