= Solution
An ideal <conductor in electrostatic equilibrium> has zero field in its interior and constant potential on its surface. Grounding fixes that potential to zero. The tangential field is continuous and therefore vanishes; the normal field has the jump fixed by the induced <surface charge density> $\sigma$:
$$
\boxed{E_x=E_y=0\quad(z=0^+),\qquad E_z(0^+)=\frac{\sigma}{\varepsilon_0},\qquad \mathbf E(0^-)=0.}
$$
Here the normal points upward out of the conductor. The normal field is not generally zero; grounding prescribes the potential, not the induced charge density.
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