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Let $C$ be the set of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. If two functions in $C$ agree everywhere except possibly on a finite set, do they in fact agree everywhere?

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  • $\begingroup$ Yes, because $\Bbb R$ is Hausdorff and finite sets have empty interior. $\endgroup$
    – user239203
    Apr 8 '21 at 11:09
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    $\begingroup$ Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$
    – Martin R
    Apr 8 '21 at 11:12
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Consider their difference if the functions disagree with each other on a finite set then the difference will be discontinuous but the difference of continuous functions is continuous .

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