Dirichlet Conditions/Examples/Reciprocal of 4 minus x squared

Example of Dirichlet Conditions
The function:
 * $f \left({x}\right) = \dfrac 1 {4 - x^2}$

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

Proof
Recall the Dirichlet conditions:

At the point $x = 2$, $\dfrac 1 {4 - x^2}$ is not defined.

Let $\epsilon \in \R_{>0}$.

Then:
 * $f \left({2 - \dfrac \epsilon 2}\right) > f \left({2 - \epsilon}\right)$

and so $x = 2$ is not a finite discontinuity.

Hence $f$ does not satisfy Dirichlet condition $(\mathrm D 3)$.