Suppose u is a harmonic function in disc $|z|<1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary?
I know v can be represented by integral use the boundary value of u, but I don't think this can help.
Can anyone prove it or give a counterexample? Thank all of you.
Try the real and imaginary parts of a conformal map from the disc to a region such as $-1/(1+y^2) < x < 1/(1+y^2)$.