SUMMARY
The function defined by the equation f(x)=(x^2-1)/(x+1) is discontinuous at x=-1 due to the denominator being zero. Although it can be simplified to f(x)=(x-1), the function is not defined at x=1, which also results in discontinuity. The definition of continuity requires that the function be defined at the point in question, and both the limit and the function value must exist and be equal at that point. Therefore, the function is continuous everywhere except at x=-1 and x=1.
PREREQUISITES
- Understanding of basic calculus concepts, particularly limits and continuity.
- Familiarity with function definitions and their properties.
- Knowledge of algebraic manipulation of rational functions.
- Experience with graphing functions and identifying discontinuities.
NEXT STEPS
- Study the definition of continuity in detail, focusing on the three criteria for continuity at a point.
- Learn about removable and non-removable discontinuities in rational functions.
- Explore the concept of limits and how they relate to continuity using examples.
- Practice graphing rational functions to identify points of discontinuity visually.
USEFUL FOR
Students studying calculus, educators teaching continuity concepts, and anyone seeking to deepen their understanding of function behavior in mathematics.