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Partial derivatives help us understand how functions change as we vary one variable while keeping others constant. For the function \(f(x, y)\), the partial derivative with respect to \(x\), denoted \(f_x(x, y)\), shows the rate of change of \(f\) when \(x\) changes, holding \(y\) constant.
In the original exercise, we find that \(f_x(0,0) = 0\), which indicates the slope of the function along the \(x\)-axis at the origin is flat. Similarly, \(f_y(0,0) = 0\) indicates the slope along the \(y\)-axis is also flat at this point.
The existence of these partial derivatives shows that the function behaves in a regular, predictable manner in the immediate neighborhood along these specific axes. However, the existence of partial derivatives alone does not guarantee full differentiability of a function at a point.

Continuity of a function at a point means you can approach the point from any path and get the same function value. For \(f(x, y)\) to be continuous at \((0,0)\), \(\lim_{(x, y) \to (0,0)} f(x, y)\) must be zero, the same as \(f(0,0)\).
In the exercise, several paths of approach to \((0,0)\) were examined. Along the paths \(x = 0\), \(y = x\), and \(y = x^2\), we see seemingly consistent limits tending toward zero at first. But continuity fails if there’s any inconsistency from different paths, leading to varying limits, making the function not continuous at this point.
Thus, while individual paths may yield limits that suggest continuity, the varying behavior along differing paths implies \(f(x, y)\) is not continuous at \((0,0)\), key for determining differentiability.

Differentiability at a point implies the function has a well-defined tangent plane or, in simpler terms, the function behaves so smoothly at the point that it can be approximated by a linear function. Typically, differentiability requires the function to be continuous and have partial derivatives at the point.
While \(f(x, y)\) does indeed have partial derivatives \(f_x(0,0) = 0\) and \(f_y(0,0) = 0\), the non-continuity at \((0,0)\), as shown by the step-by-step solution, breaks the chain of smooth behavior needed for differentiability. Essentially, if different approaches result in different behaviors, fitting a tangent plane smoothly across all paths is impossible.
This highlights why the existence of partial derivatives alone isn't a guarantee; continuity is equally crucial for ensuring differentiability in multivariable calculus.

Paths of approach are important to check how a function behaves as you move closer to a point in various ways. Different paths can reveal different limit behaviors, uncovering the function's complexity.
In the given exercise, different paths like \(x = 0\), \(y = x\), and \(y = x^2\) were used. While initially, these paths seem to suggest that the limit values are consistent, their detailed examination shows anomalies in the behavior of the function \(f(x, y)\) as these paths converge towards \((0,0)\).
Analyzing different routes helps one understand whether the function is continuous or has a consistent behavior at specific points. Ultimately, discrepancies in limit values across paths were critical in proving \(f(x, y)\) isn’t differentiable at \((0,0)\). This path-based method illustrates functions’ nuanced behaviours in multivariable calculus.