Is that a counterexample to sampling theorem?

Question

Sampling points are $\mathbb{Z}$. Sampling theorem tells us that functions with bandwidth lower than $\frac{1}{2}$ will have no aliases. Take functions with pure frequency $\frac{1}{4}$ as an example.Consider $y=sin(\frac{\pi x}{2})$ and triangle wave $$y=(-1)^k(x-2k),x\in[2k-1,2k+1],k\in\mathbb{Z}$$. Don't they coincide on sampling points? Aren't they aliases? What does sampling theorem say about that situation?

Answer 1

Your triangle wave is not band limited. It has infinite bandwidth.