  We revisit the approach to the lower critical dimension $d_{\rm lc}$ in the
Ising-like $\varphi^4$ theory within the functional renormalization group by
studying the lowest approximation levels in the derivative expansion of the
effective average action. Our goal is to assess how the latter, which provides
a generic approximation scheme valid across dimensions and found to be accurate
in $d\geq 2$, is able to capture the long-distance physics associated with the
expected proliferation of localized excitations near $d_{\rm lc}$. We show that
the convergence of the fixed-point effective potential is nonuniform when $d\to
d_{\rm lc}$ with the emergence of a boundary layer around the minimum of the
potential. This allows us to make analytical predictions for the value of the
lower critical dimension $d_{\rm lc}$ and for the behavior of the critical
temperature as $d\to d_{\rm lc}$, which are both found in fair agreement with
the known results. This confirms the versatility of the theoretical approach.
