Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the...

Continue reading...# real analysis

## Rationals Dense but Sparse

Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the...

Continue reading...## Devil’s Staircase

Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically...

Continue reading...## Space-filling Curves

Consider a square in the plane. Is it possible to draw a curve in the square that touches every point inside the...

Continue reading...## Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to...

Continue reading...## Multidimensional Newton’s Method

You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did...

Continue reading...## Cantor Diagonalization

We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as...

Continue reading...## Continuous but Nowhere Differentiable

You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have...

Continue reading...## Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...

Continue reading...## Sierpinski-Mazurkiewicz Paradox

If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces...

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