Some time ago Xi'an asked What is the equivalent for cdfs of MCMC for pdfs? The naive answer would be to use "approximate" Metropolis algorithm in form

Given $X^{(t)} = x^{(t)}$

1. generate $Y \sim q(y|x^{(t)})$
2. take $$ X^{(t+1)} = \begin{cases} Y & \text{ with probability } & \min\left( \frac{F(Y+\varepsilon) - F(Y-\varepsilon)}{F(x^{(t)}+\varepsilon) - F(x^{(t)}-\varepsilon)} , 1 \right)\\ x^{(t)} & \text{ otherwise.} \end{cases} $$

where $F$ is a target CDF and $\varepsilon$ is some small constant. This enables us to use Metropolis algorithm with CDF's.

The question is: is there any reason why this may actually be a bad idea?