An example for the negation of the reflection rule in complex analysis/geometry is the Oka principle, often informally expressed as ``whatever can be done continuously (on Stein manifolds), can be done holomorphically" The starting point is the following theorem from 1939 by K. Oka: The second Cousin problem on a domain of holomorphy can be solved by holomorphic functions if it can be solved by continuous functions.

This negates the rule (or so I think), because holomorphic objects are more complicated than continuous ones.

More applications of Oka principle can be found in the survey by Forstneric and Larusson, located here:

http://nyjm.albany.edu/j/2011/17a-2v.pdf

and in G. Elencwajg's answer to the MO question

Most helpful heuristic?

The answers to that question can be probably mined for more examples/counterexamples.

An example for the negation of the reflection rule in complex analysis/geometry is the Oka principle, often informally expressed as ``whatever can be done continuously (on Stein manifolds), can be done holomorphically" The starting point is the following theorem from 1939 by K. Oka: The second Cousin problem on a domain of holomorphy can be solved by holomorphic functions if it can be solved by continuous functions.

This negates the rule (or so I think), because holomorphic objects are more complicated than continuous ones.

More applications of Oka principle can be found in the survey by Forstneric and Larusson, located here:

http://nyjm.albany.edu/j/2011/17a-2v.pdf

and in G. Elencwajg's answer to the MO question

Most helpful heuristic?

The answers to that question can be probably mined for more examples/counterexamples.