Let $f , g : \mathbb { C } \longrightarrow \mathbb { C }$ be complex analytic, and let $h : [ 0,1 ] \longrightarrow \mathbb { C }$ be a non-constant continuous map. Suppose $f ( z ) = g ( z )$ for every $z \in \operatorname { Im } h$, then $f = g$. (Here $\operatorname { Im } h$ denotes the image of the function $h$.)