Why the optimizing scalar becomes a function T(x)

In f-divergence the density ratio $\frac{p(x)}{q(x)}$ depends on the location x. Therefore the optimizing scalar t can also depend on x. Write $t=T(x)$, where T maps the data space X to the real numbers. $T(x_1)$, $T(x_2)$, and $T(x_3)$ may be different, but they need not be.

The optimization variable is now an entire function T, not one fixed number. A function class $\mathcal{T}$ is a collection of candidate functions. Restricting this class may make optimization more practical, but the supremum over a restricted class cannot exceed that over the full admissible class.