I'm uncertain on how to proceed with this question:
Given a continuous real valued function $f$ defined on $[0,1]$, show that the function $$F(t)=\int_0^1f(x)e^{tx}dx$$ is analytic over $\mathbb{R}$
I thought it may be useful to show that the Lagrange remainder goes to zero in a neighborhood of every $t_0$, but it is not clear to me how to bring this forward. Thanks
I will give you a couple of hints on how to tackle this, you can fill in the details or ask me in the comments if you have any doubt.
First, (rigorously) use differentiation under the integral sign (Leibniz's Rule) in order to compute the derivatives of $F$ (and to show that they exist, at the same time). Then, expand $e^{tx}$ as a power series. Finally, use the Dominated Convergence Theorem in order to prove that $F$ is indeed equal to its Taylor Series.