real analysis - How can I show that a r.v. with cumulative distribution is continuous?

I want to show that, if $F_X$ is the cumulative distribution function of a random variable $X$, then $X$ is absolutely continuous iff $F_X \in C^1(\mathbb{R})$ ?

I know absolutely continuous means that there exists $f(x) \ge 0, \int_{\mathbb{R}} f(x) dx = 1$ such that $P(X \in A) = \int_A f(x) dx$, and this happens if and only if $F_X (x) = \int_{-\infty} ^ x f(t)dt$

Now, we know $F_X (x) = \int_{-\infty}^x F'_X (t)dt$, and since $F_X \in C^1$, we know that $F_X'$ exists, it must be the density and $X$ is continuous.

I am interested in a rigorous proof. Can this be considered rigorous?

Can some hypothesis be relaxed?