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\begin{center}
\textbf{Partial Differentiation}

\textit{\textbf{Limits}}
\end{center}

\textbf{Question}

Given a function $f(x,y)$ and a point in its domain $(a,b)$. Assume
that the single variable functions $g$ and $h$ are described as
\begin{eqnarray*}
g(x) & = & f(x,b)\\
h(y) & = & f(a,y)
\end{eqnarray*}
If $g$ is continuous at $x=a$ and $h$ is continuous at $y=b$, does
this mean that $f$ is continuous at $(a,b)$?

Also, does continuity of $f$ at $(a,b)$ mean that $g$ is continuous at
$a$ and that $h$ is continuous at $b$. Justify your answers?


\textbf{Answer}

Let
$f(x,y) = \left \{ \begin{array}{llrl} \ds \frac{2xy}{x^2+y^2} & \ \
\textrm{if} & \ \ (x,y) \ne & (0,0)\\
0 & \ \ \textrm{if} & \ \ (x,y) = & (0,0)
\end{array} \right.$

Let $a=b=0$. If $g(x)=f(x,0)$ and $h(y)=f(0,y)$, then $g(x)=0 \
\forall x$, and $h(y)=0 \ \forall y$.

So $g$ and $h$ are continuous at $0$. However $f$ is not continuous.

If $f(x,y)$ is continuous at $(a,b)$, then $g(x)=f(x,b)$ is continuous
at $x=a$ as
$$\lim_{x\to a} g(x) = \lim_{x\to a, \ y\to b} f(x,y) = f(a,b).$$

Similarly, $h(y)=f(a,y)$ is continuous at $y=b$.

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