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nikolany
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Is there a function f:R-->R that is continuous at π and discontinuous at all other numbers?
Thx
Thx
Is there a Continuous Function f:R-->R Discontinuous at All Other Numbers?
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SUMMARY
The discussion centers on the existence of a continuous function f: ℝ → ℝ that is continuous at π and discontinuous at all other numbers. The proposed function f(x) = 0 for irrational x and f(x) = x for rational x was initially considered, but a more suitable function is f(x) = (x - π)¹⁸ for rational x and 0 otherwise. This latter function successfully meets the criteria of continuity at π while being discontinuous at all other points.
PREREQUISITES- Understanding of real-valued functions
- Knowledge of continuity and discontinuity in mathematical analysis
- Familiarity with rational and irrational numbers
- Basic comprehension of polynomial functions
- Study the properties of continuous functions in real analysis
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- Investigate the implications of discontinuities in mathematical functions
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Mathematicians, students of real analysis, and anyone interested in the properties of continuous and discontinuous functions.