Suppose I have a set $S$ of functions $\mathbb{R}\to\mathbb{R}$ that is continuous with respect to some functional norm (suppose $L^\infty$, if it matters).
Given a point $f \in S$, suppose I want the set of all "directions" pointing to the interior of $S$ at $f$. That is, the set: $$T_f = \left\{\frac{g - f}{\|g - f\|} \;\vert\; g {\mathrm{\ in\ the\ punctured\ neighbourhood\ of\ } f}\right\}$$
If $S$ is a differentiable manifold, this would be the tangent space of $S$ at $f$.
Is there a more general term for $T_f$ when $S$ isn't known to be a differentiable manifold? Does $S$ have to be a differentiable manifold? What are the necessary conditions on $S$ for this kind of construction to make sense?
Context: I've got a cost functional $J$ that accepts a function $u:\mathbb{R}\to\mathbb{R}$ in a function space $\mathbb{K}$, and I'm looking for a standard name/notation for the set of directions that one can "move" at $u$ while still remaining in $\mathbb{K}$. Assume $J$ is a differentiable function on $\mathbb{K}$, hence the Lie derivative of $J$ would be well-defined along all these directions.