Let $E$ be a regular hexagon centered at the origin of $\mathbb{R}^2$. Let $f$ be the harmonic function in $E$ with boundary value 1 on one of the sides of $E$ and boundary value $0$ on each of the remaining sides. What is the value of $f$ at the origin?
This question has shown up on an old PDE qual I am studying. This problem is causing me a lot of concern, because $f$ seems to be discontinuous on the boundary of $E$. But, given that $f$ is harmonic (and thus continuous) in $E$, shouldn't the boundary values of $f$ also define a continuous function?
Hints or explanations are greatly appreciated!