SUMMARY
The function f(x) defined as f(x) = sin(x)/x for x≠0 and f(0) = 1 is continuous and differentiable for all x. To prove continuity at x=0, the limit of f(x) as x approaches 0 must equal f(0), which is established using the limit definition. The derivative f'(x) is also continuous, as it can be shown that the difference quotient approaches the derivative at x=0, confirming differentiability.
PREREQUISITES
- Understanding of limits and continuity in calculus
- Knowledge of differentiability and the difference quotient
- Familiarity with piecewise functions
- Basic trigonometric functions and their properties
NEXT STEPS
- Study the limit definition of continuity in calculus
- Learn about the properties of differentiable functions
- Explore the concept of piecewise functions and their derivatives
- Investigate the application of L'Hôpital's Rule for limits involving indeterminate forms
USEFUL FOR
Students studying calculus, particularly those focusing on continuity and differentiability of functions, as well as educators seeking to explain these concepts effectively.