SUMMARY
A constant function defined on a closed interval is continuous on that interval, as demonstrated through an ε-δ proof. Given a constant function f(x) = c, the difference f(x) - f(a) equals zero for any point a within the interval. Since the absolute difference is always zero, it satisfies the ε-δ definition of continuity, confirming that for any ε > 0, there exists a δ such that |f(x) - f(a)| < ε for all x within δ of a.
PREREQUISITES
- Understanding of ε-δ definitions of continuity
- Familiarity with closed intervals in real analysis
- Basic knowledge of functions and limits
- Ability to manipulate inequalities
NEXT STEPS
- Study the ε-δ definition of continuity in depth
- Explore proofs of continuity for other types of functions, such as polynomial and rational functions
- Learn about the implications of continuity on differentiability
- Investigate the properties of closed intervals in real analysis
USEFUL FOR
Students of calculus, mathematics educators, and anyone interested in understanding the foundational concepts of continuity in real analysis.