Chapter 2.5 Exercises

A function discontinuous at every point

a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function

$$f(x)=\cases{1, \;\;x\in \mathbb{Q}\\0,\;\;x\in\mathbb{R-Q}}$$

is discontinuous at every point.

$\blacksquare$