SUMMARY
The function f(x,y) = xy / (x² + y²) is proven to be discontinuous at the point (0,0) through the examination of limits along various paths. Specifically, evaluating the function along the line y = x yields a limit of 1/2, while along the line y = cx produces a limit of c/(1+c²), demonstrating that the limit depends on the path taken. This variability confirms the discontinuity at the origin. Additionally, the function f(x,y) = (x² + y) / √(x² + y²) also does not have a limit as (x,y) approaches (0,0), as shown by different results when approaching along the x-axis and y-axis.
PREREQUISITES
- Understanding of multivariable limits
- Familiarity with continuity and discontinuity concepts in calculus
- Knowledge of evaluating limits along different paths
- Basic proficiency in algebraic manipulation of functions
NEXT STEPS
- Study the epsilon-delta definition of continuity in multivariable calculus
- Learn about polar coordinates and their application in evaluating limits
- Explore the concept of directional derivatives and their relation to continuity
- Investigate the behavior of functions with removable discontinuities
USEFUL FOR
Students of calculus, particularly those studying multivariable functions, educators teaching continuity concepts, and anyone seeking to deepen their understanding of limits in higher dimensions.