Exercise 8.4.18

Let $g_n(x)={\int }_c^{c+n}f(x,t)\mathit{dt}$. From Theorem 8.4.6, $g_n^{\prime }(x)={\int }_c^d{f}_x(x,t)\mathit{dt}$. We have $(g_n(x))\to F(x)$ pointwise, and $(g_n^{\prime })$ converges uniformly to ${\int }_c^{\infty }{f}_x(x,t)\mathit{dt}$. Therefore by the Differentiable Limit Theorem (Theorem 6.3.1),