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Math Help - Has limit everywhere but continuous nowhere?

  1. #1
    Senior Member bkarpuz's Avatar
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    Has limit everywhere but continuous nowhere?

    Hi again MHF members,

    originating from the affirmative answer of the problem "Continuous Everywhere but Differentiable Nowhere",
    I would like to ask whether or not it is possible to find a function,
    which has a limit at each point in its domain but it is not continuous at all?

    Thanks.
    bkarpuz
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    Re: Has limit everywhere but continuous nowhere?

    Yes it is possible.Let's say there is a hole at (3,3) on the function f(x)=x. There is a limit, but the function ends up not being continuous. Know the three conditions of continuity, 1. f(x) must exist 2. lim x->a f(x) must exist 3. f(x) must equal lim x->a f(x)
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    Re: Has limit everywhere but continuous nowhere?

    Quote Originally Posted by ShadowKnight8702 View Post
    Yes it is possible.Let's say there is a hole at (3,3) on the function f(x)=x. There is a limit, but the function ends up not being continuous. Know the three conditions of continuity, 1. f(x) must exist 2. lim x->a f(x) must exist 3. f(x) must equal lim x->a f(x)
    @ShadowKnight8702 you have miss-read the question. See below.
    Your example is continuous almost everywhere on its domain.

    Quote Originally Posted by bkarpuz View Post
    it is possible to find a function,
    which has a limit at each point in its domain but it is not continuous at all?
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    Re: Has limit everywhere but continuous nowhere?

    Okay, I see my error.
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    Re: Has limit everywhere but continuous nowhere?

    Quote Originally Posted by bkarpuz View Post
    originating from the affirmative answer of the problem "Continuous Everywhere but Differentiable Nowhere", I would like to ask whether or not it is possible to find a function,
    which has a limit at each point in its domain but it is not continuous at all?
    You seem to have a perverse idea as to what continuity actually means.
    Continuity is intrinsically connected to limits. Look at this webpage.
    If you will just stop and think about it, then you can see what a nonsense question this really is.
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    Re: Has limit everywhere but continuous nowhere?

    But what about the Weierstrass function Weierstrass function - Wikipedia, the free encyclopedia?

    I've always used this function as a mnemonic to remember that Differentiability implies continuity but not necessarily the other way around. Continuity everywhere, by definition, implies existence of a limit everywhere and furthermore that the limit at each point c, for all c in the domain, equals f(c).
    Last edited by Elusive1324; August 22nd 2013 at 08:48 PM.
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    Re: Has limit everywhere but continuous nowhere?

    Quote Originally Posted by Plato View Post
    You seem to have a perverse idea as to what continuity actually means.
    Continuity is intrinsically connected to limits. Look at this webpage.
    If you will just stop and think about it, then you can see what a nonsense question this really is.
    I don't follow you, Plato. If a function f is continuous then the limit of that function at c exists and is equal to f(c) for all c in the domain. However, bkarpuz is asking about a case where the limit exists everywhere but the function is not continuous. This can occur for a function whose limit everywhere in the domain exists but is not equal to what the function evaluate to at each respective point.

    Existence of a limit alone doesn't guarantee continuity.
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    Re: Has limit everywhere but continuous nowhere?

    Quote Originally Posted by bkarpuz View Post
    Hi again MHF members,

    originating from the affirmative answer of the problem "Continuous Everywhere but Differentiable Nowhere",
    I would like to ask whether or not it is possible to find a function,
    which has a limit at each point in its domain but it is not continuous at all?

    Thanks.
    bkarpuz
    Your question is inconsistent. The function you're interested in is 1. Continuous everywhere and 2. Differentiable nowhere. And then, referring to that function, you ask about a function that has the properties: 1. The limit exists everywhere. 2. Not continuous everywhere.
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    Senior Member bkarpuz's Avatar
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    Re: Has limit everywhere but continuous nowhere?

    Plato, I could do see that it does not sound good for reals. I also know that for discrete topological spaces it is not also possible. This is why I didn't say anything about the domain. It can be some kind of a dense but not complete topological space...
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