Continuity of $2$ variable function in $R^2$ when one variable function is differentiable.

Question

I have this question in an assignment and I am unable to figure it out.

"Suppose $f(x,y)$ is a function defined in $R^2$. Set $g(x) = f(x, 0)$, $h(y) = f(0, y)$. If $g$ and $h$ are differentiable at $0$ as functions in one variable does it follows that f is continuous at the origin? (If your answer is "yes", provide a proof; if your answer is "no", construct a counterexample.)"

I was able to construct a counterexample to show that it does not follow that $f$ is differentiable at the origin but I am not sure if how that relates to the continuity of $f$.

Thanks in advance.

Answer 1

I was able to construct a counterexample to show that it does not follow that $f$ is differentiable at the origin but I am not sure if how that relates to the continuity of $f$.

Consider the function defined by $$f(x,y) = \left\{ \begin{array}{cc} 1 & xy=0 \\ 0 & xy \ne 0 \end{array}\right.$$ so it takes the value $1$ on the coordinate axes and $0$ elsewhere. Clearly $g(x)=f(x,0)=1$ and $h(y)=f(0,y)=1$ so $g$ and $h$ are differentiable but $f$ obviously isn't continuous at the origin.