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\textbf{Partial Differentiation}

\textit{\textbf{Limits}}
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\textbf{Question}

Explain how the function
$$f(x,y) = \frac{x^3-y^3}{x-y}, \ \ \ (x\ne y)$$
can be defined along the line $y=x$, so that it becomes continuous at
all points of the $xy$-plane.


\textbf{Answer}

$$f(x,y) = \frac{x^3-y^3}{x-y} = x^2+xy+y^2$$
if $x\ne y$.

But $x^2+xy+y^2 =3x^2$ on the line $y=x$.

Therefore define $f(x,x)=3x^2$ and the function will equal $x^2+xy+y^2$
at all points. It will therefore be continuous at all points.

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