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[parent] function of not bounded variation (Example)

Example. We show that the function

$\displaystyle f\!:\; x\mapsto\!$   $\displaystyle \left\{ \begin {array}{ll} x\cos\frac{\pi}{x} & \mbox{when}\,\,x \neq 0,\ 0 & \mbox{when}\,\, x = 0, \end{array} \right.$  

which is continuous in the whole $ \mathbb{R}$, is not of bounded variation on any interval containing the zero.

Let us take e.g. the interval $ [0,\,a]$. Chose a positive integer $ m$ such that $ \frac{1}{m} < a$ and the partition of the interval with the points $ \frac{1}{m},\, \frac{1}{m+1},\,\frac{1}{m+2},\, \ldots,\, \frac{1}{n}$ into the subintervals $ [0,\,\frac{1}{n}],\; [\frac{1}{n},\,\frac{1}{n-1}],\;\ldots,\; [\frac{1}{m+1},\,\frac{1}{m}],\; [\frac{1}{m},\,a]$. For each positive integer $ \nu$ we have (see this)

$\displaystyle f\left(\frac{1}{\nu}\right) = \frac{1}{\nu}\cos\nu\pi = \frac{(-1)^\nu}{\nu}.$
Thus we see that the total variation of $ f$ in all partitions of $ [0,\,a]$ is at least
$\displaystyle \frac{1}{n}\!+\!\left(\frac{1}{n}\!+\!\frac{1}{n\!-\!1}\right)\!+... ...{1}{m\!+\!1}+\frac{1}{m}\right) = \frac{1}{m}+2\!\sum_{\nu=m+1}^n\frac{1}{\nu}.$
Since the harmonic series diverges, the above sum increases to $ \infty$ as $ n\to\infty$. Accordingly, the total variation must be infinite, and the function $ f$ is not of bounded variation on $ [0,\,a]$.

It is not difficult to justify that $ f$ is of bounded variation on any finite interval that does not contain 0.

Bibliography

1
E. LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset III. Toinen osa. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).



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Other names:  example of unbounded variation, function of unbounded variation

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Cross-references: contain, finite, infinite, sum, diverges, harmonic series, total variation, subintervals, points, partition, integer, positive, interval, bounded variation, continuous, function

This is version 3 of function of not bounded variation, born on 2008-03-24, modified 2008-03-25.
Object id is 10438, canonical name is FunctionOfNotBoundedVariation.
Accessed 234 times total.

Classification:
AMS MSC: 26A45 (Real functions :: Functions of one variable :: Functions of bounded variation, generalizations)
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