My question is concerned about a recent theorem that I saw in class, but rahter to understand it completely I had a few questions.
The theorem goes like this: Be f:$\Bbb R^n \rightarrow \Bbb R$ and $a \in \Bbb R$ Such as exist $r>0$ and also exist the partial derivatives in $B_r(a)$ and they are continuous in a then f is differentiable in a
But when I ask my teacher what if we only ask to exist every single directional derivative and to be continuous in a, he told me that my idea was not correct because we could make a function such as (in $\Bbb R^3$) the cut made by the planes along the direction of the derivative in the surface where continuous, but if we take another plane it gave us another continuous function but completele different from the other cut. The difference, he told me, lied in something rewarding the analysis in the function in the theorem such as the different cuts along the axes forced the funtion to behave smothly and continuous in different direction.
But heres where it lies my problem, I dont see in the theorem where it lies that idea, what should add into my idea to accept differentiability? Someone help me understand what makes this theorem acceptable in different directions to make f differentiable in a?