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Continuity is defined at a point while uniform continuity is defined on a set. If a function $f : X \to Y$ is continuous at every point $x \in X$, then does this imply it is uniformly continuous on the set $X$?

I think this should be the case. Because if it is continuous at each point, then for each point, we can find a $\delta > 0$ for a given $\epsilon > 0$. Can't we just take the supremum of all $\delta$'s to get a single $\delta > 0$ for each $\epsilon$.

See also Definition

rims
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1 Answers1

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Let $(X,\tau)$ and $(Y,\rho)$ denote topological spaces and let $f:X\to Y$ be a function.

Then the following statements are equivalent:

  • function $f$ is continuous.

  • function $f$ is continuous at every point $x\in X$.

Here $f$ is by definition continuous if preimages under $f$ of elements of $\rho$ are elements of $\tau$.

Here $f$ is by definition continuous at $x$ if for every $V\in\rho$ that satisfies $f(x)\in\ V$ there exists an $U\in\tau$ such that $x\in U$ and $f(U)\subseteq V$.

drhab
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