As in the title: I am interested in why it is necessary to exclude the constant sequence from the following definition of a functional limit:
Note the $0 < |x-c|$. Why would the definition break down if only $|x-c|<y\delta$ was required?
2026-05-14 16:51:28.1778777488
Why exclude the constant sequence from this definition
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If you allow for the constant sequence, then the existence of the limit implies the continuity of $f$ at $c$ and $L = f(c)$. Using the above definition, you can also handle discontinuous functions.
Edit: As mentioned by Daniel Fischer above, it would be even equivalent to the continuity of $f$.