Discussion
Let \(f: \mathbb{R}^{2} \to \mathbb{R}\) be a function defined by
\[ f(x,y) = \begin{cases} \dfrac{xy}{|x| + y}, & y \ne -|x| \\ 0, & \text{otherwise.} \end{cases} \]
Then which one of the following statement is TRUE?
Answer: A
The true statement is \(f\) is NOT continuous at (0,0).
Partial derivatives first. On the \(x\)-axis, \(f(x,0) = 0\), and on the \(y\)-axis, \(f(0,y) = 0\), so both difference quotients are 0 and \(f_{x}(0,0) = f_{y}(0,0) = 0\). That rules out every option giving a partial derivative the value 1.
Now approach the origin along a path close to the excluded curve \(y = -|x|\), where the denominator vanishes. Take \(x \gt 0\) and \(y = -x + x^{2}\):
\[ \begin{aligned} |x| + y &= x^{2} \\ xy &= -x^{2} + x^{3} \\ f &= -1 + x \to -1 \end{aligned} \]
Along the axes \(f \to 0\), but along this path \(f \to -1\), so the limit at (0,0) does not exist and \(f\) is not continuous there, even though both partial derivatives exist.
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