I know that the fourier transform $\hat f$ of a $L^1$ function $f$ is (even uniformly) continuous by estimating $\lvert \hat f(x+ h) - \hat f(x)|$ and using DCT.
Now given a function $f\in C_c^\infty(\mathbb R^n)$ (smooth and compactly supported in the ball $\{\lvert x \rvert \leq R\}$) and $\zeta \in \mathbb C^n$, the expression $$\hat f(\zeta) := (2\pi)^{-n/2} \int_{\mathbb R^n} e^{-i\zeta\cdot x} f(x) \, dx$$ is well-defined. Is there a similar way to show that this function is (uniformly?) continuous? I ultimately want to prove analyticity by Osgood's lemma but I need continuity first. So far I couldn't apply the technique used for the real case since I end up with something like$$\lvert \hat f(\zeta) - \hat f(\zeta + h)| \leq (2\pi)^{-n/2}\int\lvert f(x) \rvert e^{R \cdot\text{ Im } \zeta}\lvert1-e^{-ih\cdot x} \rvert dx.$$