Poisson kernel

1. May 24, 2010

jem05

im proving this theorem that says if u is a cont fct (piecewise) we have the fct
U(z) = 1/2*pi /integral (0 --> 2*pi) P_z(t) u(e^i*theta)d(theta) is harmonic, |z|<1
where P_z(t) is the poisson kernel.
the theorem says that at a point of discontiniuty, we have a jump M-m
and the fct U(*theta) - (M-m)/*pi p(x,y) or U(*theta) + (M-m)/*pi p(x,y) is cont
for p(x,y) = (tan)^-1 (y-b)/(x-a) which coincides with the angle the line joining (a,b) the pt of discont. and (x,y) forms with the x-axis
i was able to prove this.
then they say the result follows easily that as you approach the point of discont from within the circle, U(*theta) can take any value l between m and M. U can be expressed as a linear combination of M and m.
i need help seeing this,...
thank you for any hints