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? (Headbang)

Thanks.

**bkarpuz**

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)

Re: Has limit everywhere but continuous nowhere?

Quote:

Originally Posted by

**ShadowKnight8702** 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** it is possible to find a function,

**which has a limit at each point in its domain but it is **__not continuous at all__?

Re: Has limit everywhere but continuous nowhere?

Re: Has limit everywhere but continuous nowhere?

Quote:

Originally Posted by

**bkarpuz** 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.

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).

Re: Has limit everywhere but continuous nowhere?

Quote:

Originally Posted by

**Plato** 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.

Re: Has limit everywhere but continuous nowhere?

Quote:

Originally Posted by

**bkarpuz** 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? (Headbang)

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.

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...