How can you prove the existence of a nonzero function from the subset $U= \{z| 0 \leq Re z \leq 1\}$ of $\mathbb C$ to $\mathbb C$ which is holomorphic on the interior of $U$ and vanishes on the right boundary of $U$ ?



Existence of a special holomorphic function
If you assume that your function is continuous on this right boundary (without that, your question should not make sense), then you can use a reflexion principle to extend its real part into a harmonic function $v$ in a neighbourhood of $z=1$. This harmonic function is the real part of a holomorphic function, thus your holomorphic function does extend across this boundary, as a holomorphic function $f$. This function has non-isolated zeroes, thus $f\equiv0$.


Thus there is no solution to your problem.


