There are many academic sources, books and articles, introducing forward interest rate curve. For example, those authors define $f(\tau)=f(\tau;\beta_0,\beta_1,\beta_2,\lambda)$ as a function of time to maturity $\tau$, dependent on parameters to be estimated. Such approach is use while introducing, e.g., Nelson-Siegel or Svensson model.
However, given the spot interst rate structure $R(\tau)$, the forward rate $f(\tau)$, estimated right now, should also have one more argument $t$, i.e. $f(\tau)$ is in fact $f(t;\tau)$, since it's the rate for the period $[t;t+\tau]$, implied from the spot term structure $R$.
Please, let me know, if I'm missing something. Where is $t$ in the forward rate curve definition?