Suppose that $f$ is an integrable function on the interval $[a, b]$, we define $g(x) = \int_{a}^{x} f(y) dy $, prove that $g(x)$ is continuous on interval $(a, b)$.
I first thought of rewriting the definition of $g(x)$ as an abstract integral: $$g(x) = \int_{[a, x]} f d\lambda $$, where $\lambda$ is the Lebesgue measure and then try to show that $|g(x) - g(c)| < \epsilon$, when $|x-c| < \delta(\epsilon)$. (Which I think I can do)
The main doubt is : Is this conversion to an abstract integral valid?