why I have this discontinuity when I plot the following function by using the MATLAB?

I used Matlab to plot the phase of this function $f_k(z)= (-0.1540 + 0.2600i)+ ( 0.4347 + 0.0914i)z+( 0.7756 - 0.4566i)z^2.$, but its figure shows discontinuity as shown in the following figure. Could anyone please from these community experts explain why this discontinuity happened?

I will appriciate any help.

many thanks

[Dyuman Joshi]

"why I have this discontinuity when I plot the following function by using the MATLAB?"

Because that's how the function behaves

%Function
f_k = @(z) (-0.1540 + 0.2600i)+ ( 0.4347 + 0.0914i)*z+(  0.7756 - 0.4566i)*z.^2;

%Phase of the function
fphase = @(z) angle(f_k(z));

vec = linspace(-6,6,200);
[X,Y] = meshgrid(vec);
Z = X+i*Y;

surf(X,Y,fphase(Z))
view(2)

[Torsten]

Reminder from one year ago:

https://uk.mathworks.com/matlabcentral/answers/1796340-where-the-jump-of-the-phase-function-happen?s_tid=srchtitle

The function angle(p(z)) is discontinuous where p(z) is real-valued and <= 0.

By solving p(z) = x for z (x real-valued) and restricting x to non-positive values in the plot, we find the curves where angle(p(z)) is discontinuous.

syms z x
p = (-0.1540 + 0.2600*1i)+ ( 0.4347 + 0.0914*1i)*z+(  0.7756 - 0.4566*1i)*z.^2;
assume(x,'real')
ps = p - x;
sol = solve(ps==0,z);
rsol1 = matlabFunction(real(sol(1)));
isol1 = matlabFunction(imag(sol(1)));
rsol2 = matlabFunction(real(sol(2)));
isol2 = matlabFunction(imag(sol(2)));
x = -10:0.1:0;
hold on
plot(rsol1(x),isol1(x),'o')
plot(rsol2(x),isol2(x),'x')
hold off
grid on

[Aisha Mohamed]

Yes, Sorry for that I know you all provided me with a very good explanation,

but because I read this discontinuity happens because the tan function is a multi-valued function and I find myself with a confious issue.

[Aisha Mohamed]

I understand your explanation that this discontinuity happens at the value where f_k(z) is real and negative. My question is why the discontinuity happens at these points?

Many thanks for all help

[Torsten]

but because I read this discontinuity happens because the tan function is a multi-valued function and I find myself with a confious issue.

I'd also be confused if I would have read this. It's not the "tan" function, but the "atan2" function that has this discontinuity:

https://uk.mathworks.com/help/matlab/ref/angle.html

https://uk.mathworks.com/help/matlab/ref/atan2.html

[Aisha Mohamed]

Thank you very much

[Torsten]

My question is why the discontinuity happens at these points?

Because atan2 is discontinous for those z values where f_k(z) is real and non-positive.