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Question

# Respond with true or false to the following assertion. Be prepared to defend your answer.If $f_x(0,0)=f_y(0,0)$, then $f(x, y)$ is continuous at the origin.

Solution

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The given statement is false.

Let's consider the following function:

\begin{aligned} f(x,y)&=\dfrac{xy}{x^2+y^2}\\ f(0,0)&=0 \end{aligned}

This function is not continuous at the origin, however, its partial derivatives at the origin do exist and they are equal to zero.

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