An example of a function $f: \mathbb{R}^2 \to \mathbb{R}$ that is $C^1$ (i.e., continuously differentiable) but not $C^2$ (i.e., its second derivative is not continuous) can be constructed using a function that has continuous first derivatives but a discontinuous second derivative.

Here is a classic example:

Example:

Define the function $f: \mathbb{R}^2 \to \mathbb{R}$ as:

\sqrt{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ #### Analysis: 1. **First derivative (continuity):** - For $$(x, y) \neq (0, 0)$$, $$f(x, y) = \sqrt{x^2 + y^2}$$, which is a smooth function. Therefore, the partial derivatives of $$f$$ exist and are continuous away from the origin. - At $$(0, 0)$$, the first partial derivatives of $$f$$ are: $$\frac{\partial f}{\partial x}(0, 0) = 0 \quad \text{and} \quad \frac{\partial f}{\partial y}(0, 0) = 0$$ These derivatives can be verified using the definition of the derivative, and they are continuous at the origin. 2. **Second derivative (discontinuity):** - The second partial derivatives, such as $$\frac{\partial^2 f}{\partial x^2}(x, y)$$ and $$\frac{\partial^2 f}{\partial y^2}(x, y)$$, do not exist at $$(0, 0)$$ in a continuous manner. For instance, the second derivative in the direction of $$x$$ near $$(0, 0)$$ tends to infinity as $$(x, y)$$ approaches the origin. Specifically, at the origin, there is a sharp change in curvature, causing the second derivatives to be undefined or discontinuous. This means that while the first derivatives are continuous, the second derivatives are not. ### Conclusion: This function is **$$C^1$$** because the first derivatives are continuous everywhere, but it is **not $$C^2$$** because the second derivatives are discontinuous at the origin. --- Would you like a more detailed explanation of how the second derivatives behave near the origin? Or perhaps another example? ### Related Questions: 1. What does it mean for a function to be of class $$C^k$$? 2. How can we determine if a function is $$C^2$$ at a specific point? 3. What is the significance of the existence of higher-order derivatives in multivariable calculus? 4. Can a function be continuous but not differentiable? Provide an example. 5. How does the concept of continuity relate to differentiability in multivariable calculus? ### Tip: When analyzing the differentiability of multivariable functions, always check the continuity of partial derivatives first, as the existence of continuous partial derivatives is a necessary condition for higher-order differentiability.