A simple example of a function $f : \mathbb{R}^2 \to \mathbb{R}$ that belongs to class $C^1$ but not to class $C^2$ is:

$f(x, y) = |x| \cdot y$

This function is $C^1$ in $\mathbb{R}^2$ (i.e., it is continuous and has continuous first partial derivatives), but it is not $C^2$ because its second partial derivatives are not continuous at $x = 0$. Specifically, the mixed partial derivative $\frac{\partial^2 f}{\partial x \partial y}$ is not continuous at $x = 0$.

Let me know if you'd like more explanation or if you have any other questions!

Follow-up questions:

1. Can you explain the difference between $C^1$ and $C^2$ functions in more detail?
2. How can you check whether a function is continuous or differentiable in $\mathbb{R}^2$?
3. How do second-order partial derivatives impact the classification of a function as $C^2$?
4. Can the function $f(x, y) = |x| \cdot y$ be modified to be $C^2$?
5. What are some typical examples of $C^2$ functions in multivariable calculus?

Tip: In multivariable calculus, checking whether the function is continuously differentiable at the boundary (such as at $x = 0$) is often the key to determining whether it's in $C^2$.