  We study the local magnetization in the 2-D Ising model at its critical
temperature on a semi-infinite cylinder geometry, and with a nonzero magnetic
field $h$ applied at the circular boundary of circumference $\beta$. This model
is equivalent to the semi-infinite quantum critical 1-D transverse field Ising
model at temperature $T \propto \beta^{-1}$, with a symmetry-breaking field
$\propto h$ applied at the point boundary. Using conformal field theory methods
we obtain the full scaling function for the local magnetization analytically in
the continuum limit, thereby refining the previous results of Leclair, Lesage
and Saleur in Ref. \onlinecite{Leclair}. The validity of our result as the
continuum limit of the 1-D lattice model is confirmed numerically, exploiting a
modified Jordan-Wigner representation. Applications of the result are
discussed.
