Let $R$ be the ring of germs of continuous functions $\mathbb R \rightarrow \mathbb R$. It is clear that this is a local ring with maximal ideal $\mathcal m$ consisting of those germs $f$ with $f(0)=0$.
What is not clear to me is why $(\mathcal m)^2 =\mathcal m$ should hold. Perhaps someone can give me a small hint how to show this.