  Two currently debated problems in galaxy evolution, the fundamentally local
or global nature of the main sequence of star formation and the evolution of
the mass-size relation of star forming galaxies (SFGs), are shown to be
intimately related to each other. As a preliminary step, a growth function $g$
is defined, which quantifies the differential change in half-mass radius per
unit increase in stellar mass ($g = d \log R_{1/2}/d \log M_\star$) due to star
formation. A general derivation shows that $g = K \Delta(sSFR)/sSFR$, meaning
that $g$ is proportional to the relative difference in specific star formation
rate between the outer and inner half of a galaxy, with $K$ a dimensionless
structural factor for which handy expressions are provided. As an application,
it is shown that galaxies obeying a fundamentally local main sequence also
obey, to a good approximation, $g \simeq \gamma n$, where $\gamma$ is the slope
of the normalized local main sequence ($sSFR \propto \Sigma_\star^{-\gamma}$)
and $n$ the Sersic index. An exact expression is also provided. Quantitatively,
a fundamentally local main sequence is consistent with SFGs growing along a
stationary mass-size relation, but inconsistent with the continuation at $z=0$
of evolutionary laws derived at higher $z$. This demonstrates that either the
main sequence is not fundamentally local, or the mass-size relation of SFGs has
converged to an equilibrium state some finite time in the past, or both.
