How to plot real roots of complex polynomials avoiding branch cuts?

Question

Lets say I have the following matrix:

M={{1,3+E^(I x),-1+2I x},{3+E^(-I x),2,E^(3I x)},{-1-2I x,E^(-3I x),-2}};

$$ M=\begin{pmatrix} 1& 3+\mathrm e^{ix} & -1+2ix\\3+\mathrm e^{-ix}&2&\mathrm e^{3ix}\\-1-2ix&\mathrm e^{-3ix}&-2\end{pmatrix} $$

The matrix, being Hermitian, has real eigenvalues. If I plot them, I get the following graph:

Eigenvalues[M]
Plot[%, {x, -1, 1}]

As you can see, we get discontinuous curves at $x\approx 0.51$, I believe, because of the branch cuts of the roots of the characteristic polynomial (ie, the roots are something like $A(x)\pm B(x)$ and the sign depends on $x$, so the roots change $A(x)+B(x)\to A(x)-B(x)$ at some $x\approx 0.51$). My question is: how can I solve this issue?

Please, consider the following aspects in regard to possible answers:

1) The curves need not be drawn in different colors: they can all be, for example blue.

2) The actual matrix I have is much more complicated that $M$, and it depends on a lot of parameters, so a definite localisation of the discontinuities has to be found at the time of plotting (no exact algebra is possible).

My attepmt

Draw the curves at specific points:

Eigenvalues[M]
ListPlot[Transpose@Table[%, {x, -1, 1, .05}],PlotStyle->Gray]

This is not really convincing, because Id rather have a continuous curve, as with Plot. Joined->True doesnt help. There has to be a better way.

ADDENDUM

As @J.M. said, this can be fixed by defining a function:

m[x_]:=Eigenvalues[M]

The problem is, this doesnt work for more complex situations; consider, for example, an extension of the matrix above:

M2={{1,3+E^(I x),-1+2I x,E^(I x)},{3+E^(-I x),2,Exp[3I x],1},{-1-2I x, E^(-3I x),-2,3},{E^(-I x),1,3,4}};

$$ M_2=\begin{pmatrix} 1&1+3\mathrm e^{ix} & -1+2ix & \mathrm e^{ix}\\ 3+\mathrm e^{-ix} &2&\mathrm e^{3ix}&1\\ -1-2ix & \mathrm e^{-3ix}&-2&3\\ \mathrm e^{-ix}&1&3&4\end{pmatrix} $$

so that

m2[x_]:=Eigenvalues[M2]
Plot[m2[x], {x, -1, 1}]

the same behaviour as before.

Answer 1

Quick fix:

Plot[Sort@m2[x], {x, -1, 1}]

Answer 2

I'm kicking myself for not thinking of ybeltukov's answer, but here's mine. It's another (less efficient) way of sorting, but this has the advantage of preserving colorings.

pts = Table[m2[x], {x, -1., 1, 1/32.}];

Do[
  pts[[i]] = MinimalBy[
    Permutations[pts[[i]]], 
    Norm[Abs[pts[[i - 1]] - #]] &
  ][[1]], 
  {i, 2, Length[pts]}
];

ListLinePlot[Transpose[pts], DataRange -> {-1, 1}]

Answer 3

I'm not sure whether this works on more complex situations, but if you just define it as a function, it appears to give what you are looking for:

m[x_] := Eigenvalues[M]
Plot[m[x], {x, -1, 1}]