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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.

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  • $\begingroup$ 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? $\endgroup$ – Jack M Feb 24 '15 at 21:54
  • $\begingroup$ Obvious is a relative term. $\endgroup$ – Mathemagician1234 Feb 24 '15 at 22:15
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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.

On the other hand, there are numerous examples where you begin with a continuous functions and directional derivatives existing in all sorts of directions and yet... the function is not continuously differentiable. The interesting thing is that continuity of partial derivatives suffices to piece together the tangent plane. To my taste, this is the thing students might find surprising: directional derivatives are not enough. You need directional derivatives which are continuously assigned near the point of tangency. Of course, if we appreciated the difficulty of finding limits in the multivariate case, then this difficulty is familiar. It is the same trouble as it contacts the definition of the differential in terms of the Frechet limit.

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