= Solution
Take a unit-face <zero-coupon bond> with $P(T,T)=1$. Its continuously compounded <zero-coupon yield to maturity>, <instantaneous forward rate>, and <short rate> are
$$
\boxed{R(t,T)=-\frac{\log P(t,T)}{T-t},\qquad
f(t,T)=-\partial_T\log P(t,T),\qquad
r(t)=\lim_{T\downarrow t}R(t,T)=f(t,t).}
$$
The last equality assumes the appropriate maturity differentiability at the diagonal. Integration of the forward-rate definition and differentiation of the yield definition give
$$
P(t,T)=\exp\left[-\int_t^Tf(t,u)du\right]=e^{-(T-t)R(t,T)},\qquad
\boxed{R(t,T)=\frac1{T-t}\int_t^Tf(t,u)du,\quad f(t,T)=R(t,T)+(T-t)R_T(t,T).}
$$
At fixed observation time $t$, the <yield curve> plots this yield against maturity $T$; it describes today's discount prices at different horizons. Forward rates are extracted from these prices, not identified with the subsequent realized path of the <short rate>.
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