Solution
= Solution
Apply the preceding interval formula on a short interval and use the <change of variables> $v=x+c$:
$$
P(u\leq U\leq u+du)=\int_{u-c}^{u+du-c}f(x)\,dx
=\int_u^{u+du}f(v-c)\,dv.
$$
Comparison with the defining integral for a <probability density> gives
$$
\boxed{t(u)=f(u-c)}.
$$
At continuity points, this can also be read as $P(u\leq U\leq u+du)=f(u-c)\,du+o(du)$.