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Determining continuity of regions/curves from inequalities

Is there an easy way in Mathematica to determine whether the region or curve 
formed by a system of inequalities is continuous or not?

For example, the output of some function (e.g. Reduce) might be as follows:

x>2 && y>0

which forms a continuous region. Again, the following output

(x<2 && y<0) || (x>2 && y>0)

is not continuous. Similarly, for curves.

Given such a system of inequalities, how to determine whether the 
region/curve it forms is continuous or not? Or in other words, if I pick any 
two random points, say P1 and P2, lying on the output curve/region, does 
there exist a continuous path lying entirely within the output curve/region 
from P1 to P2?

Any help will be appreciated.


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