Suppose $a$ and $b$ are two vectors in $R^3$ and field-wise continuous at point $x_0$. It is possible to have $a.b$ discontinuous at $x_0$?. If yes, could you cite any example.
2026-03-25 11:16:12.1774437372
Is it possible to have dot product of two continuous vector fields to be discontinuous at a point?
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No, it is not possible.
If the fields are continuous at $x_0$, their components are continuous there. But their dot product is the sum of products of their components, and multiplication and addition of continuous functions preserves continuity.