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daolian89
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Homework Statement
Where is the function f(x) continuous?
f(x) =
x, if x is irrational
0, if x is rational
Continuity of Dirichlet looking function
- Thread starter daolian89
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SUMMARY
The function f(x) defined as f(x) = x for irrational x and f(x) = 0 for rational x is continuous only at irrational points. The function outputs 0 for all rational inputs, indicating that rational points are points of discontinuity. Therefore, the function is continuous at all irrational numbers and discontinuous at all rational numbers. This conclusion is based on the behavior of the function at different types of inputs.
PREREQUISITES- Understanding of continuity in real analysis
- Knowledge of rational and irrational numbers
- Familiarity with piecewise functions
- Basic concepts of limits and discontinuities
- Study the properties of continuous functions in real analysis
- Explore the concept of limits and their relation to continuity
- Learn about piecewise function behavior and analysis
- Investigate examples of functions with similar discontinuities
Students of mathematics, particularly those studying real analysis, and educators looking for examples of continuity and discontinuity in functions.