I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the directional derivatives to exist at a point, but for the function to not even be continuous at that point.
How then, do we define a derivative that in a topological vector space that implies continuity and differentiability? Is it best to turn to the definition of Frechet derivatives? what about a Frechet derivative, that is, defining it as a linear operator with higher order mappings being $k$-linear mappings, that makes this notion uniform? I'm currently working with a space where the quotient $$ \frac{f(a+h)-f(a)}{h}$$ has no meaning, so I must either define the derivatives as directional derivatives (bad) or find another way.