Properties of distribution function

For a continuous random variable $X$, the cdf satisfies six standing properties, closely paralleling — but not identical to — the discrete case:

1. $0\le F(x)\le1$ for every $x$.
2. $F(x)$ is real-valued and non-decreasing: $x<y\Rightarrow F(x)\le F(y)$.
3. $F(x)$ is continuous everywhere (unlike the discrete step-function cdf, which jumps at each support point — a continuous $X$'s cdf never jumps, precisely because no single point carries probability).
4. $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to\infty}F(x)=1$.
5. $P(X>x_1)=1-P(X\le x_1)=1-F(x_1)$. …