= Solution
Writing $z=x+iy$ and $f=u+iv$, the <Cauchy-Riemann equations> are
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
With continuous second derivatives, mixed derivatives commute. Thus
$$
u_{xx}+u_{yy}=v_{yx}-v_{xy}=0,\qquad v_{xx}+v_{yy}=-u_{yx}+u_{xy}=0.
$$
Both real and imaginary parts are <harmonic>.
For $g(z)=\operatorname{Re}z$, the real and imaginary parts are $x$ and zero, so $u_x=1\ne0=v_y$. For $h(z)=\bar z$, they are $x$ and $-y$, so $u_x=1\ne-1=v_y$. The necessary <Cauchy-Riemann equations> fail at every point. Hence \b[both functions are nowhere <holomorphic>], indeed nowhere <complex differentiable>.
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