Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\mathbb{C}$. Is it possible to extend $f$ to a complex-valued **continuous** function on $X\cup\partial X$?

In particular, I am interested in the case where $X$ is the Cayley graph of a discrete hyperbolic group.

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