= Solution
For a nonnegative continuous event time $T$, the <survivor function> is $F(t)=\mathbb P(T>t)$, and its <probability density function> is
$$
\boxed{f(t)=-F'(t).}
$$
The <hazard function> is the instantaneous event rate conditional on being event-free at that time:
$$
\boxed{h(t)=\lim_{\epsilon\downarrow0}\frac{\mathbb P(t\le T<t+\epsilon\mid T\ge t)}{\epsilon}=\frac{f(t)}{F(t)}\quad(F(t)>0).}
$$
If the event is certain to occur in finite time, the density is proper: $\int_0^\infty f(t)\,dt=1$, so $F(t)\to0$. Since $F(0)=1$ and $F'=-hF$, the <cumulative hazard function> is $H(t)=-\log F(t)$. Therefore \b[a certain eventual event requires unbounded integrated hazard]:
$$
\boxed{\lim_{t\to\infty}\int_0^t h(u)\,du=+\infty.}
$$
If the lifetime has a finite upper endpoint, the same divergence occurs as that endpoint is approached, and $H$ is interpreted as infinite beyond it.
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