The Family-Vicsek scaling of island-size distributions is a well-known phenomenon in epitaxial growth of surface islands. Analytic scaling functions obtained from rate equations for surface growth at small critical size rarely fit the results of kinetic Monte Carlo (KMC) simulations, mainly due to uncertainty in the scaled capture number C(x) in the Bartelt-Evans theory. We have recently presented a refinement of the theory that yields double-exponential scaling function. This function fits well the KMC results for compact islands, but fails to reproduce the data for point islands. Here we further develop this approach by showing that analytic Family-Vicsek scaling requires a certain form of C(x) that eliminates discontinuities at large sizes. We present a family of empirical power-law, exponential, and double-exponential functions that describe a transition from the plateau-like behavior of C(x) at small x to linear increase at large x and yield analytic scaling with the required normalizations. The scaling function obtained with double-exponential transition function fits well the KMC data on both scaled capture numbers and island-size distributions.