The answer I gave in your last question addressed this issue in part (at least for linear scales). For an arbitrary scale, you can just modify the example
% the given colormap from the last question
cmap = [0.851 0.851 0.851;0.498 1 1;0.298 0.8 1;0 0.6196 1;0 0 0.9529;0.498 1 0.498;0.09804 1 0.09804;0.09804 0.6 0.09804;1 0.749 1;1 0.4 1;0.8 0.09804 1;1 1 0;1 0.651 0;0.9529 0.298 0;0.8 0 0];
% dummy data
x = linspace(0,1,100);
y = x.';
z = max(x,y);
% plot setup
contourf(x,y,z,size(cmap,1)-1)
axis equal
colormap(cmap)
% wrangle the discrete colorbar tick alignment
cb = colorbar;
cbscalein = cb.Limits;
cbscaleout = [min(z(:)) max(z(:))];
ticks = linspace(cbscaleout(1),cbscaleout(2),size(cmap,1)+1);
cb.Ticks = diff(cbscalein)*(ticks-cbscaleout(1))/diff(cbscaleout) + cbscalein(1);
cb.TickLabels = [0:0.1:1 5:5:25]; % this is your arbitrary set of tick values
