# Tag Archives: Continuum Hypothesis Trilogy

## Continuum Hypothesis III

Suppose you pick randomly a real number. What is the probability that it equals to 1? The probability is zero. Suppose $$X\subset [0,1]$$ is a countable subset of the unit interval. What is the probability that a randomly picked real … Continue reading

Differentiability of Space Filling Curves A Peano curve is a surjective (onto) function $$f\colon\mathbb{R}\to\mathbb{R}^2$$. Apparently such an f cannot be smooth. To see this consider the restrictions of this function to closed intervals $$f\restriction [n,n+1]$$. By smoothness and compactness the … Continue reading