Directionally derivative $\lim_{t\searrow 0}\frac{(x+th)(\omega)-x(\omega)}{t}$ for $x\in C(\Omega)$

Question

Let $C(\Omega)$ be the space of continuous functions over a compact metric space $\Omega$. We fix $\omega \in \Omega$, what is the directionally differentiable at $x\in C(\Omega)$? More specifically, what is the meaning of direction $h$ when we talk about $h$ in function space, i.e. $x+th \in C(\Omega),\forall t\in [0,1]$. According to the definition of directionally differentiable, we have $\lim_{t\searrow 0}\frac{(x+th)(\omega)-x(\omega)}{t}$ exist. What is the limit? My conjecture is $h(\omega)$.