Get the set of points from constant function

Question

I have a function f defined by points:

finput = {{0, 1/5}, {1/5, 1/5}, {1/2, 1}, {1, 0}};
ListLinePlot[finput, PlotRange -> {{0, 1}, {0, 1}}]

Then I create a piecewise linear function from it

pcw[pts_, otherwise_: - 1] :=
  Module[{ptsPairs, line, x},
    line[p_] :=
      Module[{p1 = p[1], p2 = p[[2]], m, b}, 
        m = (p2[[2]] - p1[[2]])/(p2[1] - p1[1]); 
        b = p1[[2]] - p1[1] m; 
        {mx + b, p1[1] <= x <= p2[1]}]; 
    ptsPairs = Partition[pts, 2, 1]; 
    ptsPairs = Select[ptsPairs, #[1][1] != #[[2]][1]&]; 
    Piecewise[line /@ ptsPairs /. x -> #, otherwise]&]

f = pcw[finput];

Now I would like to find patterns of the function, but the problem is to find patterns of the part, where the function is constant.

Normally I use Solve to get the set of points. But it doesn't work for this case so I am trying to use Reduce

Reduce[f[x] == 1/5, x]

But I get as result this inequality:

0 <= x <= 1/5 || x == 9/10

My question is, is there any possibility to get set of points containing these points from inequalities? It means to get as result:

{0,1/5,9/10}

Answer 1

I think an interpolation function of order 1 might be what you are looking for.

pts = {{0, 1/5}, {1/5, 1/5}, {1/2, 1}, {1, 0}};
f = Interpolation[pts, InterpolationOrder -> 1];
Plot[f[x], {x, 0, 1}]

Then

f /@ Range[0, 1, 1/10]

{1/5, 1/5, 1/5, 7/15, 11/15, 1, 4/5, 3/5, 2/5, 1/5, 0}

Answer 2

Try

Cases[  0 <= x <= 1/5 || x == 9/10   , z_?NumericQ -> z, Infinity]
(*{0, 1/5, 9/10}*)

to get the numbers in your condition.