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To prove that the function \( f(x) \), defined as \[ f(x) = \begin{cases} 2, & \text{if } x \text{ is irrational}, \\ -2, & \text{if } x \text{ is rational}, \end{cases} \] is discontinuous using the sequential definition of continuity, we follow these steps: ### Sequential Definition of Continuity: A function \( f \) is continuous at a point \( c \) if for every sequence \( \{x_n\} \) that converges to \( c \), the sequence \( \{f(x_n)\} \) converges to \( f(c) \). In other words: \[ \lim_{n \to \infty} f(x_n) = f(c) \] ### Step 1: Choose a point \( c \in \mathbb{R} ______ _________ ____ _____ _______ ________.
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