EDB — 19Y

[19Y]Say which of these functions $f:[0,1]\to ℝ$ are Riemann integrable:

1. the characteristic function of the Cantor set.

2. $f(0)=0$, $f(x)=\sin (1/x)$

3. $f(0)=0$ and

   $$f(x)=\frac{1-\cos (x)}{x^2+|\sin (1/x)|}$$

4. $f(x)=0$ if $x$ is irrational, $f(x)=\cos (1/m)$ if $x=n/m$ with $n,m$ coprime. [ [19Z]↺↻]

5. $f(x)=0$ if $x$ is irrational, $f(x)=\sin (1/m)$ if $x=n/m$ with $n,m$ coprime.