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Let f be defined on the interval [-5, 5] by
f(x) = x when x is rational,
and f(x) = -x when x is irrational.
Which of the following is true about the continuity of f?
- f is continuous at every point except x = 0
- f is discontinuous at every point except x = 0
- f is continuous at all points
- f is discontinuous at all points
Correct answer: f is discontinuous at every point except x = 0
Solution
The function f is defined differently for rational and irrational numbers, leading to discontinuity at all points except x = 0, where both definitions converge to the same value. At x = 0, both f(0) = 0 and the limit as x approaches 0 from both rational and irrational sides is also 0, making it continuous only at that point.
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