real analysis - Continuity/differentiability at a point and in some neighbourhood of the point

For a function $f: U \to \mathbb{R}$ where $U$ is a subset of $\mathbb{R}$, it seems like that it being continuous at a point doesn't imply that there is a neighbourhood of the point where it can be continuous. Similarly, it seems like that it being differentiable at a point doesn't imply that there is a neighbourhood of the point where it can be differentiable. I was wondering if there are some counterexamples to confirm the above?

Added:

What are some necessary and/or sufficient conditions for continuity/differentiability at a point and in some neighbourhood of the point to be equivalent?

Can the case of continuity be generalized to mappings between topological spaces?

Thanks and regards!