Discussion
Suppose that f is a function on R2 defined by :
\[ f(x, y) = \begin{cases} \dfrac{xy}{x^{2} + y^{2}}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} \]
Then, which of the following statements is/are true ?
S1 : f is continuous at (0, 0).
S2 : Both partial derivatives exist at (0, 0).
Answer: C
Only S2 is true.
Along the line \(y = mx\), \(f(x, mx) = \frac{m}{1 + m^{2}}\), which depends on m: it is 0 along the axes and \(\frac{1}{2}\) along \(y = x\). The limit at the origin does not exist, so f is not continuous there and S1 is false.
The partial derivatives use only the axes, where f is identically 0:
\[ f_{x}(0, 0) = \lim_{h \to 0} \frac{f(h, 0) - f(0, 0)}{h} = 0 \]
and likewise \(f_{y}(0, 0) = 0\). So S2 holds. This is the standard example showing that existence of partial derivatives does not imply continuity in two variables.
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