SUMMARY
The function f(x) = 1/(|x|+1) is proven to be uniformly continuous on R using an epsilon-delta proof. The proof begins by letting ε > 0 and aims to find a δ such that for all x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε. The discussion emphasizes the need for algebraic manipulation to identify an appropriate δ that satisfies the uniform continuity condition.
PREREQUISITES
- Epsilon-delta definition of continuity
- Understanding of real-valued functions
- Basic algebraic manipulation skills
- Knowledge of limits and continuity concepts
NEXT STEPS
- Study the epsilon-delta definition of uniform continuity in detail
- Explore examples of uniform continuity proofs for different functions
- Learn about the implications of uniform continuity in analysis
- Investigate the relationship between uniform continuity and differentiability
USEFUL FOR
Students in calculus or real analysis, educators teaching continuity concepts, and mathematicians interested in function properties and proofs.