Background: I forgot all my knowledge of harmonic functions in partial differential equations classes and calculus classes. Now all my knowledge of harmonic functions is in complex analysis classes.
Definition: So far that I've seen harmonic is defined on a non-singleton set, usually a domain/region/open set of $\mathbb C$ (or Riemann surface or complex manifold). My guess is that this is to later say that holomorphic (eg on region $G$) implies harmonic (on $G$).
You can (sensibly?) define harmonic at a point (aka a singleton set) right?
1.1. What I understand is like you can indeed sensibly define terms like continuous, real differentiable and complex differentiable and even holomorphic at a point.
- 1.1.1. In the case of holomorphic, it's simply that the point's neighbours will also be points of holomorphicity.
1.2. But you can't quite sensibly define terms like uniform continuity at a point because there's some vacuousness to make the function automatically uniformly continuous. I don't think you can even define uniform continuity at a point in the holomorphic at a point kinda way.
- 1.2.1. I think it's the same with real or complex analytic...or maybe analytic is in the first category too.)
1.3. I think harmonic is in the first category.
If no: why not? If yes: I guess complex differentiable a point doesn't imply harmonic at a point right? (and I guess harmonic at a point doesn't imply real analytic at a point...etc etc)
Why I think yes for harmonic:
It's just like $U_{xx}(x_0,y_0)+U_{yy}(x_0,y_0)=0$ (for $U: A \to \mathbb R$ with $x_0 \in A \subseteq \mathbb R^2$). I believe we can even talk about like $f'(x_0)=g'(x_0)$ (for $f,g: A \to \mathbb R$ with $x_0 \in A \subseteq \mathbb R$).
But why I guess no for harmonic (based on comment):
I guess you can satisfy the Laplace equation at a point, but I think it's about the other conditions.