Understanding and Applying Dirichlet's Conditions for |x| on [-\pi, \pi]

[Suvadip]

How to show in details that the function |x| satisfies Dirichlet's conditions on \(\displaystyle [-\pi \pi]\).;

I know the Dirichlet's conditions, but facing problems to apply it on the given function.

 

[ThePerfectHacker]

How to show in details that the function |x| satisfies Dirichlet's conditions on \(\displaystyle [-\pi \pi]\).;

I know the Dirichlet's conditions, but facing problems to apply it on the given function.

Dirichlet's condition, I assume means what most books called piecewise $\mathcal{C}^1$.

Let $f:[a,b]\to \mathbb{R}$ be a function. Here is the definition of what it means for $f$ to be piecewise $\mathcal{C}^1$:

(i) Let $p\in (a,b)$, $\lim_{x\to p^+} f(x) \text{ and }\lim_{x\to p^-} f(x)$ both exist as finite numbers. We denote these numbers by $f(p+)$ and $f(p-)$ respectively and call them the left and right hand limits.

(ii) For $p=a,b$, the endpoints, we require that $f(a+)$ and $f(b-)$ to exist.

(iii) For every $p\in (a,b)$ we require that,
$$ f'(p+) = \lim_{x\to p^+} \frac{f(x) - f(p+)}{x-p} \text{ and }f'(p-) = \lim_{x\to p^-} \frac{f(x)-f(p-)}{x-p} $$
To both exist as finite numbers, we call these the right-hand and left-hand derivatives.

(iv) For $p=a,b$ we require for $f'(a+)$ and $f'(b-)$ to exist in the way defined above.

Which of these conditions can you verify?