An application of Open Mapping theorem

While studying analytic function i came across a problem which is:

If $D$ is the unit disc in $\mathbb C$ and $f:\mathbb C→D$ analytic with $f(10)=\frac{1}{2}$, then what is $f(10+i)$?

I think it is an application of Open mapping theorem which states that analytic functions which are non-constant map open sets to open sets."

My confusion is:By the only given condition $f(10)=\frac{1}{2}$, how can I deduce that $f$ is constant?. Is it not possible for a non- constant analytic function to attain a constant value at any point in its domain?