Indefinite Integral $F(x)=\int_{-\infty}^xf(t)dm(t)$ Proof Check

Question

I have the following proposition and proof, and I was just wondering if anyone could tell me if I was correct in my reasoning or not.

Let $f$ be a non negative integrable function. Show that the function $F$ defined by $F(x)=\int_{-\infty}^xf(t)dm(t)$ is continuous, where $m$ denotes the Lebesgue measure.

Proof: Since $f$ is integrable, we have that $\int_{-\infty}^{\infty}f(t)dm(t)=K$ for some $K \in \mathbb{R}$, and this gives us that $lim_{x\rightarrow \infty} \int_{-\infty}^xf(t)dm(t)=K$. From this, given $\epsilon > 0$, there exists an $N$ such that for all $x_1,x_2 \geq N$, we have that $K-\int_{-\infty}^{x_1}f(t)dm(t)<\frac{\epsilon}{2}$ and $K-\int_{-\infty}^{x_2}f(t)dm(t)<\frac{\epsilon}{2}$. So we have that $|F(x_1)-F(x_2)|=|\int_{-\infty}^{x_1}f(t)dm(t)-\int_{-\infty}^{x_2}f(t)dm(t)|$, and this is $\leq |K-\int_{-\infty}^{x_1}f(t)dm(t)|+$ $|K-\int_{-\infty}^{x_2}f(t)dm(t)|<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$.

I feel like all the components of the proof are there but it still feels a bit dodgy to me. What am I missing, and where are my errors in logic?

Answer 1

Fix $x_0$. We have:

$$ |F(x) - F(x_0)| \le \int_{x_0}^x |f(t)| \, dm(t) $$

Since $f$ is integrable, $|f|dm$ is bounded and absolutely continuous with respect to $m$. Hence, for each $\varepsilon > 0$, we can find $\delta > 0$ so that:

$$ |x - x_0| < \delta \Rightarrow \int_{x_0}^x |f(t)| \, dm(t) < \varepsilon $$