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I keep finding proofs discussing how subsets of continuous functions are continuous, but I’m puzzled because what if my continuous function is, for example, y=x, and I make my subset of x the even integers. Then as far as I can tell I would have just created a discontinuous subset from a continuous function.

Can someone clarify this for me?

Caleb Huber
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    What is a discontinuous set? – José Carlos Santos Aug 27 '22 at 19:29
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    I think the standard phrase is "restrictions of continuous functions", rather than "subsets of continuous functions". (unless I have misunderstood your scenario.) – 311411 Aug 27 '22 at 19:29
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    When the domain of a function is not an interval, there is a (topological) definition of continuity that's actually easier to satisfy at the "holes" or "gaps" of the domain. It becomes much different from our intuition of a continuous function (on an interval) as being one whose graph has no breaks in it. For example, every function whose domain is the even integers is continuous. – Greg Martin Aug 27 '22 at 20:14
  • Thanks! That helped make sense of what I was seeing. To clarify what I was asking about I’m going to attempt to cite the original discussion that I read below: – Caleb Huber Aug 28 '22 at 01:58
  • @MISC {1773085, TITLE = {Subsets of continuous functions are also continuous}, AUTHOR = {Ben (https://math.stackexchange.com/users/330672/ben)}, HOWPUBLISHED = {Mathematics Stack Exchange}, NOTE = {URL:https://math.stackexchange.com/q/1773085 (version: 2016-05-05)}, EPRINT = {https://math.stackexchange.com/q/1773085}, URL = {https://math.stackexchange.com/q/1773085} } – Caleb Huber Aug 28 '22 at 02:00

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