# Is it ever easier to show differentiability than continuity?

I'm TAing a course right now in multivariable calculus and in the lecture notes the professor gave the students the theorem stating that differentiability implies continuity, as well as another theorem that says a function $F(x,y)$ is differentiable at a point $(a,b)$ if $\partial_x F$ and $\partial_y F$ exist and are continuous at $(a,b)$. I was trying to think of an example where the continuity of $F(x,y)$ is not obvious, but the continuity of the partial derivatives is more apparent and came up with nothing.

• In order to even take a partial derivative, you have to show that $f$ with one variable fixed is a continuous function (in fact, that it's a differentiable function). So perhaps a good question to ask is: what kind of function is obviously continuous when one variable is fixed, but not obviously continuous in both variables? – Jack M Feb 24 '15 at 21:54
• Obvious is a relative term. – Mathemagician1234 Feb 24 '15 at 22:15

My intuition is that such an example is not possible. Let me explain. To say $\partial_x F$ and $\partial_y F$ exist continuously at $(a,b)$ is to say that $z=F(x,y)$ is a graph with a well-defined tangent plane at $(a,b,F(a,b)$. In particular, the equation of the tangent plane is: $$z= F(a,b)+\partial_x F(a,b)(x-a)+\partial_y F(a,b)(y-b).$$ This plane is the best affine approximation of $z=F(x,y)$ near the point of tangency. It follows that the function is pretty well-behaved near the tangent point as this plane is the dominant feature of the graph near enough to $(a,b,F(a,b))$. So, to me, the question is approximately restated as: "can I find a plane in which continuity of the plane is subtle at the base-point?" I think not.