SUMMARY
The function f(z) defined as f(z) = z^2/z for z ≠ 0 and f(0) = 0 satisfies the Cauchy-Riemann equations at z = 0 but is not differentiable there. The proof involves demonstrating that the limit f'(0) = lim_{h -> 0} ((f(h) - f(0))/h) does not exist. By approaching zero along different paths, it is shown that the limit yields different results, confirming the lack of differentiability at that point.
PREREQUISITES
- Understanding of complex functions and their properties
- Familiarity with the Cauchy-Riemann equations
- Knowledge of limits and continuity in complex analysis
- Basic skills in evaluating limits from different paths
NEXT STEPS
- Study the implications of the Cauchy-Riemann equations on differentiability in complex analysis
- Learn about the concept of limits in complex functions
- Explore examples of functions that satisfy Cauchy-Riemann equations but are not differentiable
- Investigate the geometric interpretation of complex differentiability
USEFUL FOR
Students of complex analysis, mathematicians exploring differentiability in complex functions, and educators teaching the properties of the Cauchy-Riemann equations.