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To prove that the function \( f(x) \) is discontinuous at each real number using the sequential definition of continuity, we follow these steps: ### Function Definition: \[ f(x) = \begin{cases} 3, & \text{if } x \text{ is irrational} \\ -3, & \text{if } x \text{ is rational} \end{cases} \] ### Sequential Definition of Continuity: A function \( f(x) \) is continuous at a point \( a \) if: \[ \lim_{x \to a} f(x) = f(a) \] This means that for every sequence \( x_n \to a \), the corresponding sequence \( f(x_n) \) must converge to \( f(a) \). In other words, for every sequence \( x_n \) that converges to \( a \), the sequence \( f(x_n) \) must converge to \( f(a) \). ________ _________ _________ __________ ___ _________ ______ ____ _____.
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