= Solution
The <short rate> is the limiting <instantaneous forward rate> at the present maturity. The continuously compounded <zero-coupon bond> price is obtained by integrating the <instantaneous forward rate> curve in its maturity variable. \b[Both requested relations are]
$$
\boxed{r_t=f(t,t),\qquad
P(t,T)=\exp\!\left(-\int_t^T f(t,u)\,du\right).}
$$
In particular $P(T,T)=1$, and $f(t,T)=-\partial_T\log P(t,T)$. The <short rate> here is instantaneous, rather than the one-period rate used in a discrete-time <bank account>.
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