SUMMARY
The function f(x) = x * sin(1/x) for x ≠ 0 and f(0) = 0 is uniformly continuous on R. The domain of f is all real numbers, and the analysis shows that for any ε > 0, there exists a δ > 0 such that if |x - y| < δ, then |f(x) - f(y)| < ε. The function is continuous across R and exhibits a bounded derivative on the interval [-1, 1], confirming its uniform continuity.
PREREQUISITES
- Understanding of uniform continuity in real analysis
- Knowledge of limits and continuity of functions
- Familiarity with the properties of trigonometric functions
- Basic calculus, including derivatives and their implications
NEXT STEPS
- Study the definition and properties of uniform continuity in detail
- Explore the behavior of bounded derivatives and their relationship to continuity
- Investigate the implications of continuity for piecewise functions
- Learn about the application of the epsilon-delta definition in real analysis
USEFUL FOR
Students of calculus, mathematicians focusing on real analysis, and educators teaching concepts of continuity and differentiability in functions.