@Markiyan Hirnyk 

You started the game asking for maths and proofs. I gave you a distilled version of the mathematical result, and you don't like it. I cannot do anything about it, I'm sorry too.

@Markiyan Hirnyk 

1. The addition of x>0 is just my method to get rid of the curve not defined at t=0.
2. These are standard notions, see wiki or any topology textbook. The theorem is elementary.
f,g are the roots (depending on t) of the quadratic equation in x (OP's f(x,t)=0).

@Markiyan Hirnyk 

OK, let's play.

1. There is only one example here.

2. The following theorem applies:
If X is a connected topological space, Y a Hausdorff space and f,g: X --> Y are continuous such that f(x)<>g(x) for x in X
then the multifunction F : X -> 2^Y, F(x) = {f(x), g(x)} has exactly two continuous selections.

Use this for X = (0,4].

@Markiyan Hirnyk 

For this example it is easy to see that there is a unique continuous curve defined in [0,4] and as shown, it can be obtained at once.
Of course, for a more general case, Preben's method should be used.

@Markiyan Hirnyk 

Please be more explicit. Are you saying that the solution plotted is not correct?
The question was to plot the curve, not to prove the unicity (I know how to do it if needed).


 

@Jjjones98 

The Fourier series in cos is the same as the Fourier series (sin&cos), the function being even in [-Pi,Pi].

Anyway, in your case:

restart;
f:=cos(alpha*x):
A:=n->1/Pi*int(f*cos(n*x),x=-Pi..Pi); # Fourier coeffs
alpha:=3/4:
f6:=A(0)/2+add(A(n)*cos(n*x),n=1..6):
numapprox:-infnorm(f-f6,x=0..Pi);

     0.5206935226e-1

You may want to compute the L^2 norm.

You cannot have two distinct Fourier series; it is unique.

@gaurav_rs 

Just change 'complex' to 'real'.

@gaurav_rs 

Probably you use an old version of Maple. In recent versions the answer is very fast.

Please post a link to such a question, I did not find one.

Your second Identity is wrong (some +/- are inverted).

@Adam Ledger 

@Yee Voon 

OK then, I just wanted to prevent a typo.

@acer 

Yes, heatmap displays the image as a background.
I don't understand why some of the PLOT objects are not documented
(even if they are not supposed to be used by the user).

You should consider the following issues about your posts:

1. Nobody wants to see a same problem posted again and again.
If it has not been answered, probably it cannot be answered
(for various reasons).

2. You use Maple 12. Very few users are able to run this version.

3. You use the document mode. This mode should be used only for presentations,
when the code works.
For testing and debugging, this would imply a conversion
and probably most users prefer to do something else with their time.

4. If you discovered a strange result produced by Maple,
it would be useful to exhibit the simplest version where
this abnormal result is still present.

@Rouben Rostamian  

Yes, because evalf/evalhf introduces some 0.*I
plot(Re(ans), ...)  solves the problem.