on the boundary of analytic functions Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| &lt; 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the boundary (e.g. $\{ e^{it}, 0 \leq t \leq \pi/2 \}$ ), is it then true that $f$ is identically zero on the entire boundary? 

I know that analytic functions that are zero at an accumulation point inside the domain of analyticity are identically zero throughout the entire domain, but I don't know what (if anything) can be said if something similar occurs on the boundary of the domain.
The only way the function can be zero on an arc of the circle is for the function to be identically zero. To see this, consider ${\displaystyle g(z) = \prod_{j = 0}^{n-1} f(e^{2\pi i j \over n}z)}$ for a large enough $n$ so that $g(z)$ is identically zero on the boundary of the disk. Then by the Cauchy integral formula, if $r &lt; 1$ then for any $w$ in the interior one has
$$g(w) = {1 \over 2\pi i}\int_{|z| = r}{g(z) \over z - w}\,dz$$
Since $g(z)$ is continuous on the compact set $\bar{D}$, $g(z)$ is bounded, so one can apply the dominated convergence theorem and let $r$ go to $1$. One obtains
$$g(w) = {1 \over 2\pi i}\int_{|z| = 1}{g(z) \over z - w}\,dz$$
$$ = 0$$
Thus $g(z)$ is identically zero. Since the zeroes of a nonzero analytic function are isolated, we conclude that $f(z)$ is identically zero as well.
It is well known that if an analytic function assumes the value zero in some set which admit an accumulation point, is identically zero: as for your question there exist also  a more general result.
Theorem (Privalov). Let $u:\Bbb D \to \Bbb C$ be an analytic function on $\Bbb D$ ($\Bbb D\equiv$ unit disk in $\Bbb C$) which is continuous on $\operatorname{clo}(\Bbb D)$. If $u=0$ in some subset of  positive Lebesgue-measure of the boundary of $\Bbb D$, then $u$ is identically zero.