Introduction:
To determine if the function f(x) = |x| is continuous, we need to analyze its behavior at different points and intervals. Continuity is a fundamental concept in calculus that refers to the absence of any sudden jumps, holes, or vertical asymptotes in a function's graph.

Analysis:
1. Continuity at x = 0:
- Let's consider the function f(x) = |x|.
- At x = 0, the value of f(x) is also 0 since the absolute value of 0 is 0.
- The left-hand limit as x approaches 0 is also 0, since f(x) approaches 0 from the left side.
- The right-hand limit as x approaches 0 is also 0, since f(x) approaches 0 from the right side.
- Therefore, the limit of f(x) as x approaches 0 exists and is equal to 0.
- Consequently, f(x) is continuous at x = 0.

2. Continuity on the positive x-axis (x > 0):
- For x > 0, the function f(x) = x since the absolute value of a positive number is the number itself.
- The function f(x) = x is continuous on the positive x-axis as it is a linear function with no jumps, holes, or vertical asymptotes.

3. Continuity on the negative x-axis (x < />
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- The function f(x) = -x is also continuous on the negative x-axis as it is a linear function without any discontinuities.

Conclusion:
Considering the analysis above, we can conclude that the function f(x) = |x| is continuous for all real numbers. It is continuous at x = 0 and also on the positive and negative x-axes. The graph of f(x) consists of two linear segments meeting at the origin without any sudden jumps, holes, or vertical asymptotes.