Sunday, 21 August 2016

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

For a function f:UR where U is a subset of 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!

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