= Solution
For any real $y$, put $k=\lfloor y\rfloor$. Since $Y$ is integer-valued,
$$
\{Y\leq y\}=\{\lfloor X\rfloor\leq k\}=\{X<k+1\}.
$$
Continuity of the <cumulative distribution function> of $X$ therefore gives
$$
\boxed{F_Y(y)=F_X(\lfloor y\rfloor+1)}.
$$
The strict inequality $X<k+1$ matters before using the absence of point masses. The resulting <probability distribution> is discrete, with <probability mass function>
$$
\boxed{\mathbb P(Y=k)=F_X(k+1)-F_X(k)\quad(k\in\mathbb Z)}.
$$
Thus \b[$Y$ has no ordinary <probability density function>] with respect to length on the real line. Its <cumulative distribution function> is a step function; its derivative is zero almost everywhere and does not recover its probabilities. The <floor function> has converted a density into masses on the integers.
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