Let \(f\) be continuous on \(\mathbb{C}\) and analytic on \(\mathbb{C} \setminus \mathbb{R}\). Prove that \(f\) is analytic on \(\mathbb{C}\).
Solution
We use Morera’s theorem. Let \(T\) be any triangle in \(\mathbb C\). If \(T\) lies entirely in one of the two half-planes, then \[ \int_{\partial T}f(z)\,dz=0 \] by Cauchy’s theorem.
In general, split \(T\) along its intersections with the real axis into finitely many polygonal pieces lying in the closed upper and lower half-planes. Approximate each real-axis edge by a parallel edge at height \(\pm\varepsilon\). Cauchy’s theorem applies to the shifted pieces, and continuity of \(f\) lets \(\varepsilon\downarrow0\). The integrals along the internal real-axis segments cancel with opposite orientations. Thus \[ \int_{\partial T}f(z)\,dz=0 \] for every triangle \(T\). Morera’s theorem now implies that \(f\) is holomorphic on all of \(\mathbb C\).