Final answer:
The effect of gravity on time dilation occurs continuously with no specific distance at which the rate does not change. This continuous variation applies even at infinitesimal distances, and would only be non-existent in a point-sized mass scenario, which is not physically possible. Near singularities, classical general relativity does not apply, and quantum gravitational effects are not fully understood.
Step-by-step explanation:
In general relativity, time dilation is a consequence of gravity. Time passes more slowly as one gets closer to a massive object. However, the question of whether there is an incremental distance at which time dilation remains the same is not a straightforward one. In theory, the effect of time dilation occurs continuously, with every infinitesimal difference in distance resulting in some level of differential time passage.
Therefore, there is no specific incremental distance at which the rate of time dilation does not change. Even at small scales, such as the distances provided in the example (1 meter vs. 1.000001 meters), general relativity predicts an infinitesimally small, but still present, difference in the passage of time.
This difference would continue to diminish as the distances become smaller and would only be non-existent in a theoretical point-sized mass scenario, which is a physical impossibility. Near singularities like those found in the center of black holes, classical general relativity breaks down, and quantum gravitational effects are expected, but not fully understood, to take over.