SUMMARY
The function $$f(x,y)=\left\{\begin{array}{ccc} (x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & , & (x,y)\neq (0,0) \\ 0 & , & (x,y)=(0,0) \end{array}\right.$$ is differentiable at the point (0,0) but exhibits discontinuous partial derivatives $$f_x$$ and $$f_y$$ at that point. Another example provided is $$f(x,y):=\frac {2xy}{x^2+ y^2}$$, which also demonstrates similar properties. Additionally, the univariate function $$f(x) := \begin{cases} x^2 \sin (1/x), &x\neq 0 \\ 0,&x=0 \end{cases}$$ serves as a classical example of a differentiable function that is not continuously differentiable.
PREREQUISITES
- Understanding of multivariable calculus concepts, specifically differentiability and continuity.
- Familiarity with partial derivatives and their properties.
- Knowledge of limits and their role in defining continuity.
- Basic proficiency in mathematical notation and functions.
NEXT STEPS
- Research the properties of discontinuous partial derivatives in multivariable functions.
- Explore the implications of differentiability in higher dimensions.
- Study examples of functions that are differentiable but not continuously differentiable.
- Learn about the application of the epsilon-delta definition of continuity in multivariable calculus.
USEFUL FOR
Mathematicians, students of calculus, and educators seeking to understand or teach the nuances of differentiability and continuity in multivariable functions.