The Lebesgue Fundamental Theorem of Calculus

Question

Let $f,g:\mathbb{R}\to\mathbb{R}$ be two Lebesgue integrable functions. If we have $$f(b)-f(a)=\int_a^bg(x)dx$$ for almost all $a,b\in \mathbb{R}$. How can we modify $f$ on a set of measure zero to make it continuous on the whole real line ?

Answer 1

Here is an outline of how to do this.

Find $a$ such that $f(b)−f(a) = \int_a^bg(x) dx$ for almost all $b$. (You need to prove that you can do so.)

Define $F(b) = f(a) + \int_a^bg(x) dx$.

By its definition $F$ matches $f$ except on a set of measure 0. Now you have to prove that $F$ is continuous.