https://www.desmos.com/calculator/bunaio1m4p
https://math.stackexchange.com/a/773960
$$ f_{n}\left(x\right)=\sum_{k=1}^{\frac{n}{2}-1}\left(\frac{2}{n}\cos\left(\frac{2k\pi}{n}x\right)\right)+\frac{1}{n}+\frac{1}{n}\cos^{2}\left(\frac{\pi n}{2}\right)\cos\left(\pi x\right) $$
$$ f_{n}\left(x\right)=\frac{1}{n}\left(\sum_{k=0}^{n-1}\left(\left(\cos\left(\pi k\right)+1\right)\cos\left(\frac{k\pi}{n}x\right)\right)+\cos^{2}\left(\frac{\pi n}{2}\right)\cos\left(\pi x\right)-1\right) $$
$$ f_{n}\left(x\right)=\frac{1}{2n}\left(\sum_{k=0}^{n-1}\cos\left(k\left(\pi+\frac{\pi}{n}x\right)\right)+\sum_{k=0}^{n-1}\cos\left(k\left(\pi-\frac{\pi}{n}x\right)\right)+2\sum_{k=0}^{n-1}\cos\left(\frac{k\pi}{n}x\right)\right)+\frac{1}{n}\cos^{2}\left(\frac{\pi n}{2}\right)\cos\left(\pi x\right)-\frac{1}{n} $$
$$ g_{n}\left(x\right)=\frac{1}{2n}\left(\frac{\sin\left(\frac{n}{2}\left(\pi+\frac{\pi}{n}x\right)\right)}{\sin\left(\frac{1}{2}\left(\pi+\frac{\pi}{n}x\right)\right)}\cos\left(\frac{\left(n-1\right)}{2}\left(\pi+\frac{\pi}{n}x\right)\right)+\frac{\sin\left(\frac{n}{2}\left(\pi-\frac{\pi}{n}x\right)\right)}{\sin\left(\frac{1}{2}\left(\pi-\frac{\pi}{n}x\right)\right)}\cos\left(\frac{\left(n-1\right)}{2}\left(\pi-\frac{\pi}{n}x\right)\right)+2\frac{\sin\left(\frac{\pi x}{2}\right)}{\sin\left(\frac{\pi x}{2n}\right)}\cos\left(\frac{\left(n-1\right)\pi x}{2n}\right)\right)+\frac{1}{n}\cos^{2}\left(\frac{\pi n}{2}\right)\cos\left(\pi x\right)-\frac{1}{n} $$
$$ g_{n}\left(x\right)=\frac{1}{n}\left(\frac{\sin\left(\frac{\pi n}{2}+\frac{\pi}{2}x\right)}{2\cos\left(\frac{\pi}{2n}x\right)}\sin\left(\frac{\pi n+\pi x-\frac{\pi}{n}x}{2}\right)+\frac{\sin\left(\frac{\pi n}{2}-\frac{\pi}{2}x\right)}{2\cos\left(\frac{\pi}{2n}x\right)}\sin\left(\frac{\pi n-\pi x+\frac{\pi}{n}x}{2}\right)+\frac{\sin\left(\frac{\pi x}{2}\right)}{\sin\left(\frac{\pi}{2n}x\right)}\cos\left(\frac{\left(n-1\right)\pi x}{2n}\right)+\cos^{2}\left(\frac{\pi n}{2}\right)\cos\left(\pi x\right)-1\right) $$
$\forall(x,n)\in\mathbb{N}^{2}$, $f_{n}\left(x\right)$ such as if $x\mod n=0$, $f_{n}\left(x\right)=1$ else $f_{n}\left(x\right)=0$.
$g_{n}\left(x\right)$ is the same as $f_{n}\left(x\right)$ but is undefined for $x=nk$ but $\lim\limits_{x \to nk}g_{n}\left(x\right)=1$.
