I try to understand the proof (the one using complex analysis) of the Müntz theorem. The first step of this long proof is to see that $f:z \mapsto \int_0^1 t^z d\mu$ is holomorphic on $\Omega : = \{z \in \mathbb C; \Re(z)>0 \}$ for a Borel complex measure $\mu$. I think I understand that this is showed using Morera and Fubini theorem. However to use Morera theorem, we need to know that $f$ is continuous. Since $|f(z)-f(z_0)| \leq \int_0^1 |t^z-t^{z_0}|d\mu$, we only need to show: \begin{equation} (1) ~~~~~\forall z_0 \in \Omega ~~ \forall \epsilon > 0 ~~ \exists \delta > 0 ~~ \forall z\in \Omega ~|z_0-z| < \delta \implies \sup_{t \in[0,1]}|t^{z_0}-t^z| < \epsilon \end{equation}
And that precisely at this point that I am lost. In this presentation (p. 34), they only say $(t,z) \to t^z$ is continuous on $[0;1] \times \Omega$ and the continuity in $t$ is uniform but I don't understand how this facts are used to prove $(1)$.
Thanks in advance!