Let $f(x,y)$ be a real-valued function defined on an open subset $U \subset \mathbb R^2$. Suppose that $f$ is twice differentiable separately in each variable and satisfies Laplace's equation $f_{xx} + f_{yy} = 0$. Using the given information, we cannot conclude that $f$ is even continuous, let alone twice continuously differentiable. Therefore, we cannot guarantee that $f$ is harmonic.
My questions are:
Suppose we know that $f$ is continuous. Is this sufficient to guarantee that $f$ is harmonic?
Suppose we know that $f$ is differentiable. Is this sufficient to guarantee that $f$ is harmonic?
Suppose we know that $f$ is continuously differentiable. Is this sufficient to guarantee that $f$ is harmonic?
Suppose we know that $f_{xy}$ and $f_{yx}$ exist. Is this sufficient to guarantee that $f$ is harmonic?
Suppose we know that $f_{xx}$ and $f_{yy}$ are continuous. Is this sufficient to guarantee that $f$ is harmonic?
Suppose we know that $f$ is twice differentiable (but not continuously so!). Is this sufficient to guarantee that $f$ is harmonic?