I'm studying for a real analysis exam and I found the following exercise:
Let $f : \mathbb{R} \to \mathbb{R}$ a non-negative and integrable function. Prove that the function $ F(x) := \int _{-\infty} ^ x f(t) dt $ is continuous.
A friend told me a way to prove it using monotone convergence theorem and the epsilon-delta definition of continuity, but it is a little complicated for me. So I think there is a simpler way to prove it using dominated convergence theorem and sequential definition of continuity:
Let $\{ x_n \}_{n \geq 1}$ be a real sequence that converges to $x$. Let define $g_n := f \cdot \mathbb{1}_{(-\infty, x_n]}$ and $g := f \cdot \mathbb{1}_{(-\infty, x]}$. Observe that $g_n$ converges pointwise to $g$ and the sequence $\{ g_n \}_{n \geq 1}$ is dominated by $f$. By dominated convergence theorem, we conclude that
$ \lim_{n\to\infty} \int _{-\infty} ^ {\infty} g_n(t) \ dt = \int _{-\infty} ^ {\infty} g(t) \ dt $.
Then, by difinition of $F$, $\{ F(x_n) \} _ {n \geq 1}$ converges to $F(x)$. This means that $F$ is continuous.
What do you think about my proof? I is it correct?