How to avoid ambiguity defining continuity / differentiability of multivariable function A function $f(x):\mathbb R^n \rightarrow\mathbb R^1$ is continuous on $\mathbb R^n\setminus \{0\}$ and the limit $\lim_{x\rightarrow0} f(x)$ exists (in a sense is bounded) in any direction to origin. 
Is this function continuous in the origin, or in order to be continuous there it has to have the same limit value there in all directions?
How to differentiate such two cases defining function continuity? 
Same question arises concerning differentiability.
I appreciate any explanations.

In order to be continuous, the limit must coincide in all directions and the function must assume that value at the origin.
However, even more is needed.
The limit must coincide regardless of the path taken towards the origin (directions correspond to straight lines).
It is poosible that the limit exist and coincide for all directions but fails to exist/coincide for some other path.
A classical example is
$$f(x,y)=\frac{x^2y}{x^4+y^2},$$
which approaches $0$ along any line but is constant and equal to $\frac12$ along the curve $y=x^2$.
For differentiability, we have the same situation.
