SUMMARY
The discussion centers on proving that a function \( f \) defined on \( \mathbb{R} \) and continuous at \( x=0 \) is also continuous at all \( x \in \mathbb{R} \) given the property \( f(x_{1}+x_{2})=f(x_{1})+f(x_{2}) \) for all \( x_{1},x_{2} \in \mathbb{R} \). Key insights include demonstrating that \( f(0) = 0 \) and utilizing the limit definition of continuity. The proof hinges on showing that \( \lim_{h\rightarrow 0} f(a+h) = f(a) \) for any \( a \), leveraging the additive property of \( f \).
PREREQUISITES
- Understanding of continuity in real analysis
- Familiarity with limit definitions of functions
- Knowledge of properties of additive functions
- Basic calculus concepts
NEXT STEPS
- Study the proof of the continuity of additive functions
- Learn about the implications of continuity at a point in real analysis
- Explore the properties of functions defined on \( \mathbb{R} \)
- Review limit theorems and their applications in calculus
USEFUL FOR
Mathematics students, particularly those studying real analysis, and educators looking for insights into function continuity proofs.
