- Suppose f is continuous in a region Ω. Prove that any two primitives of f (if they exist) difer by a constant.
Solution
Let \(F\) and \(G\) be primitives of \(f\) on the region \(\Omega\), and set \(H=F-G\). Then \[ H'(z)=F'(z)-G'(z)=f(z)-f(z)=0 \] for every \(z\in\Omega\).
Fix \(z_0\in\Omega\). Since \(\Omega\) is open, some disc \(D(z_0,r)\) lies in \(\Omega\). For any \(z\in D(z_0,r)\), the segment \[ \gamma(t)=z_0+t(z-z_0),\qquad 0\le t\le1, \] lies in the disc. The function \(t\mapsto H(\gamma(t))\) has derivative \[ H'(\gamma(t))(z-z_0)=0, \] so it is constant. Thus \(H(z)=H(z_0)\) throughout \(D(z_0,r)\). Hence \(H\) is locally constant on \(\Omega\).
A locally constant function on a connected set is constant: for any value \(c\), the level set \(H^{-1}(c)\) is both open and closed. Since a region is connected, the nonempty level set containing \(z_0\) must be all of \(\Omega\). Therefore \(F-G\) is constant on \(\Omega\).