EDB — 1T9

Exercises

1. [1T9] Note:reworked from the written exam held January 26th, 2016.

   Let \((q_ n )_{n≥1}\) be an enumeration of the rationals of \((0, 1)\) and define

   \[ f(t) {\stackrel{.}{=}}∑_{n: q_ n {\lt}t} 2^{ −n} \]

   and

   \[ g (t) {\stackrel{.}{=}}∑_{n: q_ n ≤ t} 2^{ −n} \]

   for \(t ∈ (0, 1)\).

   • Show that \(f,g\) are strictly increasing.

   • Calculate limits for \(t ↓ 0\) and \(t ↑ 1\).

   • Show that \(f\) is left continuous, \(g\) is right continuous, and that

     \[ \lim _{𝜏→ t+} f (𝜏) =g(t) \quad ,\quad \lim _{𝜏→ t-} g (𝜏) =f(t) \quad . \]

   • Also show that \(f\) is discontinuous in \(t\) if and only if \(t ∈ ℚ ∩ (0, 1)\); and similarly for \(g\).

   • What changes if we replace \(2^{ −n}\) with the term \(a_ n\) of an absolutely convergent series?

   Solution 1

   [1TB]↺↻