Maximum value of $|f(z)|$

Question

Today while Reading maximum modulus theorem I encountered this question!

Question: find the maximum value $|f(z)|$ in $|z|≤1$ for the function $f(z)=\frac{2z+1}{2z-1}$

But, how we can apply maximum modulus theorem to above function? As it is clear that, $f(z)$ is not analytic at $z= 1/2$ which lies inside $|z|≤1$ :-(

So how one can find maximum value for the given function? Please help me, stuck on this!

Answer 1

The limit does not exist at z = 1/2 because the left limit does not equal the right limit there. I think all you can say is that the function approaches its max value as z approaches 1/2 from the right.