Vanishing Zigzags of Unbounded Length

We will construct a sequence of functions on the unit interval such that it converges uniformly to zero while the arc-lengths diverge to infinity.

Kangaroo-Frog-01

Black: Frog hop. Blue: Cricket hops. Magenta: Flea hops.

Hopping Animals

Let us compare a sequence of frog hops, cricket hops and flea hops. We assume each hop is a semi-circle so that the length is easily calculated. If the total distance is {L} and there are {N} semi-circular hops, the step-size between landings is {L/N}, so each arc has length {\pi L/2N} and the total arc-length is {s= \pi L/2}, independent of {N}. Thus, the frog, cricket and flea travel equal distances.

Tent Functions

To simplify the analysis, we will replace the semi-circles by line segments. We start with the tent-function

\displaystyle f_1(x)=\begin{cases} x \,, & 0\le x \le 0.5 \\ 1-x \,, & 0.5 \le x \le 1 \end{cases}

Having defined {f_n(x)}, the next function is defined by halving the height and doubling the number of peaks in the interval {[0,1]}. The first three functions are shown in Fig. 2. Each of the functions {f_n(x)} has the same total arc-length, {\sqrt{2}}.

Kangaroo-Frog-02

Blue: f_1(x). Magenta: f_2(x). Black: f_3(x).

Sequence with Unbounded Variation

We now define another sequence, {g_n(x)} as follows. Start with {g_1(x) = f_1(x)}. At each stage, we halve the height but increase the number of peaks by a factor of 4. The first few members of the sequence are shown in Fig. 3.

Kangaroo-Frog-03

Red: g_1(x). Blue: g_2(x). Magenta: g_3(x).

There is now a big difference. At stage {n}, there are {p_n = 2^{2n-2}} peaks, each of height {h_n = 2^{-n}}. On the interval {[0,1]}, {g_n(x)} has {s_n = 2^{2n-1}} linear segments, each being longer than the height {h_n}. Therefore, the total length of {g_n(x)} over {[0,1]} is

\displaystyle \ell_n > s_n \times h_n = 2^{n-1} \,.

Thus, the total length increases without bound as {n\rightarrow\infty}. The total variation of {g_n(x)} on {[0,1]} increases without bound as {n\rightarrow\infty}. The sequence does not have bounded variation (BV).

The sequence {\{g_n(x)\}} is uniformly convergent to zero on the unit interval. This is easily seen: given {\epsilon}, choose {m} so that {h_m<\epsilon}. Then

\displaystyle |g_n(x)| < \epsilon \quad\mbox{for all\ }x\in [0,1] \mbox{\ and\ } n>m \,.

So we have a curious sequence such that {g_n(x)} tends uniformly to zero, but the length of {g_n(x)} grows without bound as {n\rightarrow\infty}.

Smooth Functions

The functions {f_n(x)} and {g_n(x)} have discontinuous derivatives. However, this is not relevant. We can construct a sequence of functions differentiable on {[0,1]} that have length growing without bound while they converge uniformly to zero. Let us set

\displaystyle F_n(x) = \frac{1}{2^{n-1}}\sin\left[\frac{\pi x}{2^{2n-1}}\right] \,.

The first four members of the sequence are shown in Fig. 4. On the basis of the arguments above, it is clear that the sequence converges uniformly to zero while the lengths of the functions grow without bound.

Kangaroo-Frog-04

First four members of the sequence {F_n(x)}.

 


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