Hi,

> @Harsh, could you brief your work in this area (finding the points
H> of 
> discontinuity and check whether a function is singular) so it will be
> clear for me where to start?

In this context the following papers by Adam Strzeboński may be very relevant:

"Real root isolation for exp–log–arctan functions" JSC V47 I3 pp282-314
"Real root isolation for exp-log functions" ISSAC 08
"Real root isolation for tame elementary functions" ISSAC 09

These papers describe (parts of) the algorithms that Mathematica uses
for some of the features the have analogue to the RoofOf for polynomials.

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