# Showing continuity of a function defined as the integral of $\chi \mapsto \chi(x)$ over the dual group $\hat G$

Let $$G$$ be a topological group. Let's assume $$G$$ is abelian, locally compact and Hausdorff. Then there exists is a Haar measure $$d \nu$$ for $$G$$ and $$d \mu$$ for $$\hat G$$. Under the assumption that the total variation of $$d \mu$$ is finite we can define

$$x \mapsto \int_{\hat G} \chi(x) d\mu(\chi)$$

as a function from $$G$$ to $$\mathbb C$$. Can we show that this defines a continuous function?