Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.

What are necessary and sufficient conditions for point evaluations at arbitrary $x \in \Omega$ to be well-defined and continuous on $F$?

**Motivation:** The Sobolev embedding theorem states that the Sobolev space $W^{k,p}(\mathbb R^n)$ embeds into the Hölder space $C^{r,\alpha}(\mathbb R^n)$ whenever $(k-r-\alpha)/n=1/p$. However, Hölder continuity is much stronger than continuity, which in turn is stronger than the existence of point evaluations. So I wonder whether those embeddings are really the most general you can get.