A function on $\mathbb R^n$ (to $\mathbb R^1$) is say $C^2$ differentiable / (continuous) on $\mathbb R^n\setminus \{0\}$.
In (the neighborhood of) point $0$ it is ($C^0$) continuous and has a finite derivative in any radial direction.
(It looks like a generalized regular cone there with its apex at point $0$.)
How to define this type of partial continuity in more precise and formal math terms?
What type of a point is this point $0$ here?
(It may seem like a kind of a generalization of a critical point of a function... Especially if we add some nice features of our function $f$ like positiveness of its radial gradient $(x\frac{\partial f}{\partial x})>0$ and maybe even its radial convexity, positiveness of its Hessian $(x\frac{\partial^{2}f}{\partial x \partial x}x)>0$, for $\forall x$ on $\mathbb R^n\setminus \{0\}$.)
Thank you very much for education.