Show that if \(f' = 0\) on a domain \(\Omega\), then \(f\) is constant on \(\Omega\)
Solution
Write \(f = u + iv\), then \(0 = 2 f' = u_x + iv_x = u_y - iu_y\), so \(\grad u = \grad v = 0\). Show \(f\) is constant along every straight line segment \(L\) by computing the directional derivative \(\grad u \cdot \vector v = 0\) along \(L\) connecting \(p, q\). Then \(u(p) = u(q) = a\) some constant, and \(v(p) = v(q) = b\), so \(f(z) = a+bi\) everywhere.