f[x_] = Piecewise[{
  {1/2 x, 0 <= x <= 2},
  {-5 x^2 + 31 x - 41, 2 <= x <= 4},
  {-3/2 x + 9, 4 <= x <= 6},
  {3 x - 18, 6 <= x <= 8},
  {-3/2 x + 18, 8 <= x <= 10}}];

g[x_] = Piecewise[{
  {f[x - 1/5], f'[x] < 0},
  {f[x + 1/5], f'[x] > 0}}];

Plot[
  {f[x], g[x]},
  {x, 0, 10},
  AxesLabel -> {x, y},
  PlotStyle -> {Dashed, Thick}]