Prove that this integral function is continuous.

If $\displaystyle f \in L^1(m) $ and $\displaystyle F(x) = \int _{- \infty}^{x}f(t)dt $, then F is continuous on $\displaystyle \mathbb {R} $

Proof so far.

Let $\displaystyle x_0 \in X $,

I need to show that $\displaystyle \forall \epsilon > 0 $, there exists $\displaystyle \delta > 0$ such that whenever $\displaystyle \mid x - x_0 \mid < \delta $ where $\displaystyle x \in X $, then I will have $\displaystyle \mid F(x)-F(x_0) \mid < \epsilon $

Now, I know have that $\displaystyle \mid F(x)-F(x_0) \mid = \mid \int _{- \infty}^{x}f(t)dt - \int _{- \infty}^{x_0}f(t)dt \mid = \mid \int _{x_0}^{x}f(t)dt \mid $

Since f is integrable, this integral exists, but how should I refine it so it is less than $\displaystyle \epsilon $ ?

Thank you.