Let $R=C(\mathbb R)$ be the ring of continuous functions $f:\mathbb R\to\mathbb C$ where the addition and the product is pointwise defined. Let $$\mathbb m_a=\{f\in R\ |\ f(a)=0\}$$ be a maximal ideal. Show when $\mathbb m_a=\mathbb m_b$ it implies that $a=b$.
Why is that true?