Let \(R=C([0,1])\) be the ring of continuous real-valued functions on the interval \([0,1]\). Let \(T \subseteq [0,1]\), and let

\(I(T) = \{f \in R : f(x) =0 \forall x \in T \}\) .

(i) Prove that \(I(T)\) is an ideal of \(R\).

(ii) If \(x \in [0,1]\) and \(m_x =I(\{x\})\), show that \(R/M_x \cong \mathbb{R}\), and hence \(M_x\) is a maximal ideal of \(R\).

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