= Uniformly small functions with nonvanishing total variation
{title2=$f_n(t)=n^{-1}\sin(2\pi nt),\quad V_0^1(f_n)=4$}
For each positive <integer> $n$, put $f_n(t)=\sin(2\pi nt)/n$ on $[0,1]$. Its <supremum norm> is exactly $1/n$, so this <sequence of functions> converges to zero by <uniform convergence>. Nevertheless,
$$
\boxed{V_0^1(f_n)=4\quad\text{for every }n\geq1.}
$$
Each of its $n$ complete <sine> oscillations contributes $4/n$: rise from zero to $1/n$, fall through zero to $-1/n$, then return to zero. Taking a <partition of an interval> through all extrema attains $4$. Every further <partition refinement> merely subdivides <monotone function> segments, so the same reasoning as in <partition example of total variation> proves the upper bound too.
\Image[/total-variation-small-amplitude-oscillations.png]
{title=The functions sin(2πnt)/n converge uniformly to zero while their total variation remains 4; all three function plots use the same vertical scale}
For a <continuously differentiable function> on a closed bounded <interval>, total variation also satisfies
$$
V_a^b(f)=\int_a^b|f'(t)|\,dt.
$$
Applied here, the <derivative> is $2\pi\cos(2\pi nt)$, and integrating its <absolute value> over $n$ periods gives $4$. The differentiability hypothesis matters: the partition definition applies much more widely than this derivative formula.<NIST Digital Library of Mathematical Functions/1 Algebraic and Analytic Methods>[NIST Digital Library of Mathematical Functions; §1.4(v), equations 1.4.33–1.4.34, Functions of Bounded Variation]{ref} Small pointwise size and small accumulated movement are different approximation goals.
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