a gallery of continuous functions
— smooth curves, unbroken paths, the quiet mathematics of analysis —
A function is continuous if, intuitively, you can draw its graph without lifting the pen from the paper. Below is a small gallery of such curves — each drawn here as a single calligraphic stroke, each marked at the points where its character is most clearly revealed.
The kintsugi gold marks repaired places. Here it marks special points: zeros, maxima, inflections — the moments where a curve briefly tells us what it is.
The sine curve breathes in and out — the heartbeat of continuity. Everywhere smooth, nowhere resting, it traces the rhythm that underlies all periodic phenomena. From the vibration of a string to the orbit of a planet, sine is the mother of all waves.
The exponential carries within it the seed of infinity — starting from quietude, it swells with ever-increasing urgency. It is the curve of compounding, of populations, of radioactive whisper. Its smoothness deceives: what begins gently soon escapes all bounds.
Where the exponential rushes, the logarithm waits. Rising swiftly at first, then slowing with the patience of stone, it maps the infinite onto the finite. The logarithm is the inverse of ambition — it grows, but learns restraint.
Polynomials are the clay of analysis — shapeable, versatile, endlessly expressive. A cubic curve dips and rises like a mountain path, each inflection point a moment of transformation. They approximate all continuous functions, the Weierstrass theorem promises, whispering that simplicity can model everything.
The Gaussian curve is nature's favorite shape — the distribution of rain, of error, of human height. Its perfect symmetry and rapid decay to zero embody a kind of mathematical humility: most things cluster near the mean, and the extraordinary is exponentially rare.
Between any two real points runs an uncountable thread. Continuity is the promise that the thread is unbroken — that small changes lead to small changes, that the world rewards careful attention.
If you have read this far, you have walked the gallery. Thank you for your slowness.