The following question was posed on an Analysis II exam:
Suppose $f: \mathbb{R}^2 \to \mathbb{C}$ is a function such that for each $x$ the function $\mathbb{R} \to \mathbb{C}: y \mapsto f(x,y)$ is integrable (as function of $y$). Find a non-trivial condition on $f$ such that: $$F: \mathbb{R}^2\to\mathbb{C}: F(x,t)=\int_{\mathbb{R}}f(x,y)e^{-2\pi iyt}dy$$ is a continuous function.
The context of this question is Lebesgue integration and (mostly 1-dimensional) Fourier transforms. I have tried using theorems for 1D Fourier transforms but I don't know how to deal with the $x$-dependence of $f(x,y)$. I have also tried a direct method using a converging sequence $((x_{n},t_{n}))_{n\in\mathbb{N}}$ with $(x_{n},t_{n}) \to (x,t)$ and then trying to show that $|F(x,t)-F(x_{n},t_{n})|\to 0$ but the calculation does not result in a clear condition on $f$. I feel like I am missing something and so any help would be greatly appreciated.
Kind regards, Jef