Let \((X,\mcm,\mu)\) be a measure space, let \(I=[a,b]\subseteq\RR\) with \(a<b\), and let \(f\colon X\times I \to \CC\) be such that \(x\mapsto f(x,t)\) is integrable for each \(t\in I\). Put \[ F(t)\coloneqq\int_X f(x, t) \dmu(x), \qquad t\in I . \]
Suppose there exists \(g\in L^1(X,\mu)\) with \(\abs{f(x, t)} \leq g(x)\) for all \(x\in X\) and \(t\in I\). If \(t_0\in I\) and \(\lim_{t\to t_0}f(x,t)=f(x,t_0)\) for every \(x\in X\), then \(\lim_{t\to t_0}F(t)=F(t_0)\). In particular, if \(t\mapsto f(x,t)\) is continuous on \(I\) for every \(x\in X\), then \(F\) is continuous on \(I\).
Suppose the partial derivative \(\frac{\partial f}{\partial t}(x,t)\) exists for all \(x\in X\) and \(t\in I\), and there exists \(g\in L^1(X,\mu)\) with \(\abs{\frac{\partial f}{\partial t}(x, t)} \leq g(x)\) for all \(x\in X\) and \(t\in I\). Then \(F\) is differentiable on \(I\) and \[ F'(t) = \int_X \frac{\partial f}{\partial t}(x, t) \dmu(x) . \]