DSolve is not able to find a solution in some points

Question

When I use the below script to solve an equation and plot the solution, the plot shows some discontinuous lines. For example at point $xapprox0.25$ the solver cannot find the answer and the plot is discontinuous. How can I fix this problem?
Clear["Global`*"]

c = -0.8;
L = 1;
b = Pi/L;

{x1, x2} = {0, L};

eqn = f[x] == c Sin[2 f[x]] + b x + Pi/2;

f[x_] = f[x] /. 
   DSolve[{D[eqn, x], 
      f[0] == (f0 /. 
         Solve[(eqn /. x -> 0 /. f[0] -> f0), f0, Reals][[1]])}, 
     f[x], {x, x1, x2}][[1]] // FullSimplify

(*InverseFunction[-(1/2) Cos[2 #1]+#1&][(11 x)/10]*)

Plot[f[x], {x, x1, x2}]

Bob Hanlon · Answer

$Version

(* "12.1.1 for Mac OS X x86 (64-bit) (June 19, 2020)" *)

Clear["Global`*"]

c = -4/5;
L = 1;
b = Pi/L;
{x1, x2} = {0, L};

eqn = f[x] == c Sin[2 f[x]] + b x + Pi/2;

The equation does not identify a unique initial condition
ic = f0 /. Solve[(eqn /. x -> 0 /. f[0] -> f0), f0, Reals]

Only one of the possible initial conditions leads to a solution
sol = Table[
   DSolve[{D[eqn, x], f[0] == ic[[n]]}, f[x], {x, x1, x2}] // 
    FullSimplify, {n, 1, 3}] // Quiet

f[x_] = f[x] /. sol[[2, 1]];

Not all values of x produce real results
f[0.1`15]

(* 1.0972751444995646 - 0.3208287706939249 I *)

Plotting,
Plot[f[x], {x, x1, x2},
 PlotRange -> {0, 5.8},
 WorkingPrecision -> 15,
 Exclusions -> All,
 PlotPoints -> 50,
 MaxRecursion -> 2]

DSolve is not able to find a solution in some points

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