Strict Inequality Using CDF

Question

Considering the definition of cumulative distribution function: $$F_{x}(x)=P[X\le x]=\int_{- \infty}^{x} f_{x}(x)dx$$ where $f_{x}$ is the probability density function of $x$, how can one obtain $P[X< x]$ ? (Note the strict inequality)

Answer 1

If the random variable $X$ has a PDF, then, for every $x$, $P(X=x)=0$ hence $P(X\lt x)=P(X\leqslant x)=F_X(x)$.