Step-by-step Solution:

Let's analyze the given function: \[ f(x) = x + |x| \] Step 1: Consider Different Cases for \( x \) \[\] The absolute value function \( |x| \) behaves differently for positive and negative values of \( x \): 1. For \( x \geq 0 \): \[ |x| = x \Rightarrow f(x) = x + x = 2x \] 2. For \( x < 0 \): \[ |x| = -x \Rightarrow f(x) = x + (-x) = 0 \] Step 2: Check Continuity at \( x = 0 \) \[\] To check continuity at \( x = 0 \), we compute the left-hand limit (LHL) and right-hand limit (RHL). \[\] Left-hand limit (as \( x \to 0^- \)): \[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} 0 = 0 \] Right-hand limit (as \( x \to 0^+ \)): \[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} 2x = 2(0) = 0 \] Function value at \( x = 0 \): \[ f(0) = 0 \] Since \( \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) \), the function is continuous at \( x = 0 \). \[\] Step 3: Conclusion Since \( f(x) \) is continuous for all real \( x \), the correct answer is: \[ {A: x \in (-\infty, \infty)} \]