Plot the staircase function (cumulative graph) of a fat Cantor set.

fat-cantor-staircase.sage

# Compute the cumulative measure function of a "fat" Cantor set (aka
# Smith-Volterra-Cantor set) with parameter alpha (meaning that
# 2^(k-1) intervals of length alpha^k are removed at the k-th stage of
# the construction.

alpha = 1/4

# Total measure of the fat Cantor set:
totmeas = N((1-3*alpha)/(1-2*alpha))

def staircase(x):
    x = N(x)
    if x <= 0: return 0
    if x >= 1: return 1
    # ycumul and yscale are the lower bound and length of the y-axis
    # interval on which we've determined the value to lie.
    yscale = N(1)
    ycumul = N(0)
    # lbnd and rbnd are the lower and upper bounds of the x-axis
    # interval on which we've located the input point x.
    lbnd = N(0)
    rbnd = N(1)
    # alphapow = alpha^(i+1) where i is how many steps in the
    # constuction we've performed (we go up to i=30).
    alphapow = N(alpha)
    for i in range(30):
        # mlbnd and mrbnd are the lower and upper bounds of the x-axis
        # interval removed in the middle ofthe interval from lbnd to
        # rbnd.
        mlbnd = (lbnd+rbnd-alphapow)/2
        mrbnd = (lbnd+rbnd+alphapow)/2
        if x<mlbnd:
            # We are left of the removed interval
            rbnd = mlbnd
            yscale /= 2
        elif x>mrbnd:
            # We are right of the removed interval
            lbnd = mrbnd
            yscale /= 2
            ycumul += yscale
        else:
            # We are in the removed interval
            return ycumul + yscale/2
        alphapow *= alpha
    return ycumul + yscale*(x-lbnd)/(rbnd-lbnd)

# For alpha=1/3 (standard Cantor set), of course, totmeas=0 so if you
# wish to plot this case, remove *totmeas below.

g = plot(lambda x: staircase(x)*totmeas, (0,1), plot_points=960)
g.save(filename="plot.png", dpi=300, aspect_ratio=1)

See https://mathoverflow.net/questions/495986/on-the-set-of-differentiability-of-a-fat-cantor-staircase and https://bsky.app/profile/gro-tsen.bsky.social/post/3lrb4fmst2c2v for what this is all about