Dirichlet Conditions/Examples/Sine of Reciprocal of x - 1

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Example of Dirichlet Conditions

The function:

$f \left({x}\right) = \sin \left({\dfrac 1 {x - 1} }\right)$

does not satisfy the Dirichlet conditions on the real interval $\left({0 \,.\,.\, 2 \pi}\right)$.


Proof

Recall the Dirichlet conditions:

$(\mathrm D 1): \quad f$ is absolutely integrable.
$(\mathrm D 2): \quad f$ has a finite number of local maxima and local minima.
$(\mathrm D 3): \quad f$ has a finite number of discontinuities, all of them finite.


Around the point $x = 1$, $f \left({x}\right)$ has an infinite number of local maxima and local minima.

Hence it does not satisfy Dirichlet condition $(\mathrm D 2)$.

$\blacksquare$


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