Dirichlet Conditions/Examples/Reciprocal of 4 minus x squared

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

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

Proof
Recall the Dirichlet conditions:

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

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

Then:
 * $\map f {2 - \dfrac \epsilon 2} > \map f {2 - \epsilon}$

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

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