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Suppose I've a function $g: I\times X\to \mathbb{C}$ where $I$ is an open interval and $I,X\subset\mathbb{R}$. Then under what conditions of $g$ is the function defined by the Lebesgue integral:

$$ f(t) \doteqdot \int_{X}g(t,x)\, d\mathcal{L}(x)$$

continuous? When is it differentiable?

I know that for each fixed $t\in I$, the single variable function $x \mapsto g(t,x) \in \mathcal{L}^1 (\mathbb{R})$.

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I guess you can read a nice solution in these notes.

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