= Solution
The required boundary values on the real axis are zero to the left of the origin and one to the right. The bounded harmonic function with those values is the <upper-half-plane harmonic measure of the positive half-axis>,
$$
\boxed{\phi(x,y)
=\frac12+\frac1\pi\arctan\frac{x}{y}
=1-\frac1\pi\arg(x+iy),\qquad y>0.}
$$
Indeed, $\arg z$ is harmonic in the upper half-plane, and the displayed function tends to $1$ on the positive half-axis and to $0$ on the negative half-axis. The uniqueness of bounded solutions of the <Dirichlet problem> identifies it with the Brownian exit probability.
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