- #1
Silviu
- 624
- 11
Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof they present in the book they begin with: "Suppose u is harmonic in G and ##z_0 ∈ G##. Let ##r > 0## such that the disk ##D[z_0, r]## is contained in G. " and as a disk is simply connected the conclusion follows from the theorem. My question is, how can you make sure that for any ##z_0 \in G## you can have a disk around ##z_0##? (For example if ##G=\mathbb{C}##\##\mathbb{R}_{<0}## and ##z_0=0##, you can't find such a disk. What am I missing?