Show that a real-valued holomorphic function is constant.
Solution
Let \(f\) be holomorphic and real-valued on a domain \(\Omega\). If \(f\) were nonconstant, the open mapping theorem would make \(f(\Omega)\) a nonempty open subset of \(\CC\) contained in \(\RR\), but \(\RR\) contains no nonempty open subset of \(\CC\). So \(f\) is constant.