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realviz — Real Analysis Visualizations

See Riemann and Lebesgue integration side by side. Measure theory for the eyes.

Real analysis is abstract until you see it. realviz draws the two great theories of integration on the same function — so the difference finally clicks.

What's the big idea?

Every calculus student learns Riemann integration first: chop the x-axis into vertical strips, add up f(xᵢ)·Δx. Lebesgue integration flips it around — chop the y-axis instead, group points by their function value, and weigh each slice by the measure of its preimage: Σ yₖ·m(Eₖ).

Riemann partitions the x-axis	Lebesgue partitions the y-axis
Same-width vertical strips	Same-height horizontal slices
Sum: Σ f(xᵢ) · Δx	Sum: Σ yₖ · m(Eₖ)
Strip color = x-position	Slice color = y-value

Same total area. Completely different philosophy.

Where Riemann breaks: the Dirichlet function

D(x) = 1 if x is rational, 0 otherwise. Every sub-interval of [0,1] contains both a rational and an irrational — so the upper sum is always 1, the lower sum always 0, and the gap never closes. Riemann gives up.

Lebesgue measures the irrationals (1) and the rationals (0), and gets a clean answer: ∫D = 0. This single function motivated a whole field.

Measure zero, yet uncountable: the Cantor set

Remove the middle third of [0, 1], then the middle third of every piece that remains, forever. Row k shows what is left after k removals; the right panel tracks the surviving measure. m(E_k) = (2/3)^k collapses to 0 exponentially — yet the limiting set has as many points as the real line.

The Cantor function is the staircase this process grows into: continuous, non-decreasing, flat almost everywhere (F' = 0 a.e.), yet it climbs from 0 to 1 — the counterexample that kills the naive fundamental theorem of calculus.

Install

pip install realviz

Quick start

from realviz import compare_integrals, riemann_plot, lebesgue_plot

# Side-by-side comparison — the main event
compare_integrals(lambda x: x**2, 0, 1, n_partitions=15)

# Or inspect each method alone
riemann_plot(lambda x: x**2, 0, 1, n_partitions=10, method="midpoint")
lebesgue_plot(lambda x: x**2, 0, 1, n_partitions=10)

# The Dirichlet function — where Riemann fails, Lebesgue succeeds
from realviz import dirichlet_illustration
dirichlet_illustration()

# The Cantor set and the devil's staircase
from realviz import cantor_set, cantor_function
cantor_set(n_levels=4)     # construction + measure decay
cantor_function()          # F' = 0 a.e., yet F(1) - F(0) = 1

Try it yourself

python examples/demo_compare.py     # x² and sin(x), side by side
python examples/demo_dirichlet.py   # the Dirichlet illustration
python examples/demo_cantor.py      # Cantor set + devil's staircase

Design principles

• Safe by default — strict input validation: functions are sampled defensively, partition counts are bounded, no eval/exec/pickle, no network, no file I/O.
• Pedagogical color — every strip is colored by what is being partitioned, so the axis-choice is visible at a glance.
• Zero magic — pure NumPy + Matplotlib, ~800 lines of core code, MIT licensed.

Roadmap

• v0.1.0 — Riemann vs Lebesgue integral comparison + Dirichlet illustration
• v0.2.0 — Cantor set construction + Cantor function (devil's staircase)
• v0.3.0 — Simple-function approximation of measurable functions
• v0.4.0 — Convergence modes (pointwise, uniform, a.e., in L¹)
• v0.5.0 — Vitali set and non-measurable sets (conceptual)

Development

git clone https://github.com/HunterZ549/realviz.git
cd realviz
pip install -e ".[dev]"
pytest               # 86 tests, CI runs Python 3.9–3.12

License

MIT — see LICENSE.