@Markiyan Hirnyk 

Unfortunately, the solution found by Maple is wrong.

(N < 0 for any epsilon>0)!

(I also started this way, but seeing that Maple has problems, I switched to asympt).

@Christian Wolinski 

You seem to be an adept of obfuscation. Why not like this (S=something)?

u:=-Omega*a*sqrt(2)*sqrt(-Omega^2*a^2-2*k*m+sqrt(Omega^2*a^2*(Omega^2*a^2+4*k*m)))/(-Omega^2*a^2+sqrt(Omega^2*a^2*(Omega^2*a^2+4*k*m))):

Q := -4*k*m+2*Omega^2*a^2*S+Omega^2*a^2*S^2:  
# S is the denominator*positive,  Q=0.

normal(simplify(u,[Q])) assuming a>0,k>0,Omega>0,m>0,S>0;
     -I

(you took S<-2 the negative root of Q just to be "more interesting").

@mskalsi 

And is it so hard to write once at the top of the blackboard:

c = a - (b-a)*f(a) / (f(b) - f(a))

?

The kernel in this example is <x-s^2, y-t^2> and not your principal ideal J.
E.g. x-s^2 is in the kernel but not in J

B:=Groebner[Basis]([x-a^2, y-b^2,s-a,t-b], plex(a,b,x,y,s,t));
remove(has,B,[a,b]);

Edit. I just have seen that you have deleted the example!!!.

@AmirHosein Sadeghimanesh 

@Kitonum 

f:= cos(2*t/m) + cos(2*(t+5)/m):
simplify(trigsubs(f)[]):
eval(%,2*t+5=m*x):
maximize(%,x=0..2*Pi);

@Carl Love 

I found a simpler one  :-)

restart;
f:= cos(2*t/m) + cos(2*(t+5)/m):
M:= maximize(eval(f, m= 7)):
subs(M=2*abs(cos(5/m)),M);

It is frustrating that maximize works only numerically (inside plot).

Not even after

F:=simplify(eval(f,t=x*m));

maximize(F, x=0..2*Pi);
or
maximize(F, x=-infinity..infinity);

produces a syntax error instead of the correct result:

@Carl Love 

You can't hope for an answer to such an elliptic question!
And what is sol?
I don't think that you will find mind readers in this forum.

@AmirHosein Sadeghimanesh 

Please provide an example.

@Markiyan Hirnyk 

For a>0, k>0 the the expression under sqrt at the numerator is <0. Now, indeed, the principal branch of sqrt is discontinuous at any z < 0, but the restriction of sqrt to (-oo, 0] is continuous, and this is enough for the continuity of our function. That is why I said that the continuity follows easily.

A variant of Joe Riel's solution also works:

L:= [1,2,3, {4,5,6}]

subsindets(L, {set,list}, op);

@Bendesarts 

@Preben Alsholm 

Probably I was wrong, it's a very old version I seem to recall.
Anyway, the main reason for the alias is to not use a(t) at all.

@Markiyan Hirnyk 

When I say easy, I really mean easy, at least for an undergraduate level.
We are both professional mathematicians, as I understand.
Anyway, I am going to answer in the future only to polite questions, i.e. without "Roly-poly toy", "empty words" etc.

restart:
alias(a=a(t)):
a - a(t);   ###  it used to work!
   a - a(t)

@John Fredsted 

You are right here. The image of the connected set (0, oo) ^ 4 by a continuous function is connected, so the function  must be constant since the only possible values are I and -I. The continuity is easy but not automatic, because the principal branch of sqrt is not continuous in C.