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We are tasked with proving that the function \( f(x) \), defined as: \[ f(x) = \begin{cases} 3, & \text{if } x \text{ is irrational} \\ -3, & \text{if } x \text{ is rational} \end{cases} \] is discontinuous at each real number using the sequential definition of continuity. ### 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 of function values \( f(x_n) \) converges to \( f(c) \). That is, if: \[ \lim_{n \to \infty} x_n = c, \] then: \[ \lim_{n \to \infty} f(x_n) = f(c). \] ### ________ _________ _______ ________ _____ _____ ______.
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