@Markiyan Hirnyk 

Unfortunately you keep producing "anti-comments". It is a high school exercise to see that the equation 
arctan(x) = x/2 has 3 real roots and -tan(_Z) + 2 _Z = 0 has infinitely many real roots.

The important fact here is that RootOf uses a non-equivalent equation for the mentioned reason.

@Kitonum 

The name FrechetDistance for the procedure is misleading; this is actually a EuclideanDistance.
It would be interesting to compute the true FrechetDistance between the paths defined by P and Q.

@nm 

This is possible only if the expression has a single variabile. And even in this case the evaluation will produce _Z again.

Try:
expr:=RootOf(f(a, _Z, 1));

A better idea would be to define `print/RootOf`  but I think It's not worth it.

@acer 

At least for a single variable solve has the possibility to return a RootOf, so, if in doubt  it should never fail :-)

Example:
solve(floor(z)=1/2);
      RootOf(2*floor(_Z)-1)

@Adam Ledger 

If you are interested only in reals then use only the branches  0 and -1
(which contain the inverses of the restrictions of y = f(x) = x * exp(x)):

#  LambertW(-1,y),   y∈[-1/e,0)       ==>  x∈(-∞,-1]
#  LambertW(0,y),    y∈[-1/e,∞)       ==>  x∈[-1,∞)

plot(LambertW(-1,y), y=-exp(-1) .. - 0.01);
plot(LambertW(0,y),  y=-exp(-1) .. 100);

@nm 

At informal level blow-up is ok. But not beyond. Have you looked at the examples above?

Some examples (for both questions) of polynomials (non-trivial but not too complicated)  would be useful such that the (possible) responder has the possibility to test the answer.

@trace 

There are >1 solutions (actually an infinity if you don't need continuity).
If an exact solution is known, then use it of course.
If not, use fsolve and restrict if needed the range for p,v,w in order to find other solutions.

@nm 

"Blow-up" is not defined.  You probably mean  limsup_(x-->a) |f(x)| = oo. But singular does not mean this (see my first comment). For example:

singular(sin(1/x),x);  # it's bounded in R
                            {x = 0}
singular(1/(1+abs(exp(1/x))), x); # it's bounded even in C
          {x = 0}, {...}

@nm 

sqrt (the principal branch used by Maple) is continuous at 0.

For singularity, it depends on the definition. In complex analysis, 0 is a non-isolated singularity.

I suspect that knowbody knows the exact definition of a singularity used in singular.
Seems to be a combination of "maths", "syntax" and "heuristics".
The singularity at infinity is even more vague.

@tomleslie  It's obviously a bug in solve. The CleanInfty procedure was designed mainly for SolveTools:-SemiAlgebraic where the simple presence of infinity produces an error. Because in SemiAlgebraic the expressions must be polynomials and the variables are real, the presence of infinity is simply not needed (e.g. x < infinity is superfluous).
In solve, a variable is considered real if it appears in an inequality; so here x<infinity could make sense if x is not a subject to other inequality; but here too one may avoid infinity using e.g. x^2 > -1.
In solve the presence of infinity does not always produce an error because the commands from SolveTools are called after some manipulations of the input.

@nm 0<0 was introduced to replace conditions like x < -infinity  or  x=infinity. Actually CleanInfty should not be used; the idea is to avoid infinity in solve because the results seem to be unpredictible.

Note. In your example the second set of r was ignored because CleanInfty was called with 3 actual parameters [it was not defined using $].

@digerdiga 

The things are a bit different, but essentially f(u) = (1-(-1)^floor(u))/(2*u^2)  was considered 0 for u>20;
(Actually  the integral was transformed with a change of variables to [0,1]  and evaluated as 0 in [0,1/20].)
You may see the exact steps executing

infolevel[`evalf/int`]:=5;

@kuwait1 

It seems that you don't read or you are not interested in the provided answers/comments.

Seems to be OK:

restart;
Digits := 15: Z := 4: r := 4:
INT := (n, i) ->
Int(-(2*r*p)^(2*i)*Z*hypergeom([3/2-n-2*i, 1/2], [3/2], Z^2/(Z^2+p^2+r^2))/((Z^2+p^2+r^2)^(1/2)*(p^2+r^2)^(n+2*i-1/2)),
     p = -2 .. 2, method = _d01ajc):
evalf(INT(6,30));
evalf(INT(6,100));
evalf(INT(6,1000));
                     -1.30904529806182 e-15  
                     -5.98504320471557 e-30   
                     -3.61141364348684 e-206   
QQ := proc(n,imax)
local i;
evalf(-2*Pi*r*(add(pochhammer((1/2)*n, i)*pochhammer((1/2)*n+1/2, i)*INT(n, i)/factorial(i)^2, i = 0 .. imax)))
end:
QQ(6,70);
QQ(6,100);
QQ(6,200);
                      0.000211915930003326
                      0.000211915930014491
                      0.000211915930014491

Actually it would be nice to be able to increase Digits, but this is complicated because non-NAG methods are too slow and need fine-tuning for this problem.

A good apprimation would be nice too but it must be done with great care.