Show that $BV[a, b]$ is not dense in $B[a, b]$ under the metric $||f||_\infty$.
I was wondering if I could get a hint.
2026-03-29 02:36:05.1774751765
Show $BV[a, b]$ is not dense in $B[a, b]$
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Suggestion: consider the function $1_\mathbb{Q}$ which takes the value 1 at every rational and 0 at every irrational. Show that if $\|f - 1_\mathbb{Q}\|_\infty < 1/2$ then $f$ has unbounded variation.