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If a function is not continuous, does it mean it has no limit? If it's false can you give me an example? It's not homework I really want to understand.

e.g., suppose $f: \mathbb{R} \to \mathbb{R}$ fails to be continuous at a point $x_0$. Then is it possible for $\lim_{x \to x_0} f(x)$ to still exist, but not equal $f(x_0)$?

Yibo Yang
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Johnna
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    It is unclear what you are asking. I assume you talk about a function $f:\mathbb{R}\rightarrow\mathbb{R}$ which is not continuous (where?) and you want to know whether or not $f$ admits a limit (at which point?). – C. Falcon Dec 24 '15 at 14:21
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    It seems clear IMO what is being asked. Can there be a discontinuous function which has a limit at some value(s). Obviously it will be continuous at a point where the limit does exist, but can it be discontinuous elsewhere. – coffeemath Dec 24 '15 at 14:25
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    @coffeemath Maybe the question is what you just said, or maybe it's asking whether the function can have a limit at a point and be discontinuous at the same point. (Which is also true, as shown in the answers.) The question allows either interpretation; is the OP even aware of this? – David K Dec 24 '15 at 14:53
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    @coffeemath By the way, in the part after "Obviously," I think you meant the function will be continuous at a point where the limit exists and is equal to the value of the function at that point. – David K Dec 24 '15 at 14:58
  • (+1) Welcome to Math.SE! If you could please re-write your question replacing each instance of "it" with a noun (as in, "If a function $f$ is not continuous at a point $a$, does that mean $f$ has no limit at $a$?", assuming that's what you want to ask), I think you'll both clarify the question in your own mind and receive more helpful answers. :) – Andrew D. Hwang Dec 24 '15 at 15:08
  • @DavidK You are correct in both your comments above. I was being hasty and was thinking function was continuous at point in question. – coffeemath Dec 24 '15 at 18:41
  • The answer to the question (regarding a particular point $x_0$) depends on how we define the limiting value of the function at that point, i.e., "deleted" v.s. "non deleted". If $f$ has a non-deleted limit at $x_0 \in \text{Dom}(f)$, then this necessarily implies continuity of $f$ at $x_0$. See Exercise 3.1.1 of Terrance Tao's Analysis II, or this question: https://math.stackexchange.com/questions/1794652/deleted-versus-non-deleted-limits – Yibo Yang Jan 23 '23 at 19:14

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No, a function can be discontinuous and have a limit. The limit is precisely the continuation that can make it continuous.

Let $$f(x)=1\text{ for }x=0,\\f(x)=0\text{ for }x\ne0.$$

This function is obviously discontinuous at $x=0$ as it has the limit $0$.

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Consider first the function $I(x)$ which is $1$ on rationals and $0$ on irrationals. If this is multiplied by a function such as $(x-a)$ then the result will be discontinuous everywhere but at $a$ yet the limit as $x \to a$ will exist and be $0$.

coffeemath
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Imagine function $f$ which has the same value $x_n$ on $[n,n+1)$. If $(x_n)$ has a limit, then $\lim_{x \to + \infty}f(x)$ also exists.

More subtle example: let $f(x)=1^x$, $f(0)=0$. $\lim_{x \to 0}f(x)$ exists, even though $f$ isn't continuous in $0$.

Also, for $sign(x)$, $\lim_{x \to +0}f(x)$ exists (as well as $\lim_{x \to -0}f(x)$).

The only bound between continuity and having a limit is that function continuous in point has a limit in that point (which equals its value), not the other way around.

Abstraction
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