= Solution
The <cumulative distribution function> is \b[$F_X(x)=\mathbb P(X\le x)$]. In the absolutely continuous setting of a <probability density function>, the <probability density function> is a nonnegative integrable function $f_X$ satisfying
$$
\boxed{F_X(x)=\int_{-\infty}^x f_X(t)\,dt,\qquad\int_{-\infty}^{\infty}f_X(t)\,dt=1.}
$$
Thus $\mathbb P(a<X\le b)=\int_a^b f_X(t)\,dt$ and $F_X'=f_X$ almost everywhere. A merely continuous <cumulative distribution function> need not possess a <probability density function>; the density assumption means absolute continuity.
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