Let $P$ and $Q$ be two probability distributions on $\mathbb{R}$. The goal is to obtain a notion of ``distance'' between $P$ and $Q$, e.g., total variation distance, K-L divergence.

Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a test function. If $f$ satisfies $$ \mathbb{E}_P [ f] = \mathbb{E}_Q[ f],$$ that is, $f$ has the same expectation under $P$ and $Q$. Moreover, if $$ \text{Var}_P[f] = 0~~\text{and}~~ \text{Var}_Q [f] >0,$$ can we say that $TV(P, Q) =1$?

For example, let $P $ be the Rademacher distribution and $Q $ be the standard normal distribution. Consider $f(x) = x^2$. Then $$ \mathbb{E}_P [ f] = \mathbb{E}_Q[ f] =1, ~~\text{Var}_P[f] =0, ~~\text{Var}_Q[f] =1. $$ In this case $TV(P,Q) =1$.

As an extension, if we have $$ \text{Var}_P[f]\leq \epsilon \text{and}~~\text{Var}_Q[f] \geq C $$ for some small number $\epsilon$ and a constant $C$, can we say something about $TV(P,Q)$?