On the proof of a theorem in real analysis The following text is from the book Advanced Calculus by Patrick M. Fitzpatrick : 

 

There are two things that I don't understand with the proof : 

1- In order that the last equation (last line) to hold, the proof must also show that $f(x)$ is continuous but it doesn't. The sequence ${\{f_n}\}$ to be pointwise convergent to $f$ doesn't guarantee that $f$ is contentious on $I$. How to show that $f$ is contentious on $I$?

2- A part of the first claim of the two-claim of the theorem is $f$ is differentiable but it doesn't prove that. How to show that $f$ is differentiable on $I$?
1- $g$ is a uniform limit of continuous functions, therefore it is continuous.

2- $f$ is the integral of a continuous function (see point 1), so it is differentiable.
1- $g$ is a uniform limit of continuous functions, therefore it is continuous.

2- $f$ is the integral of a continuous function (see point 1), so it is differentiable.