Thread: Continuous functions in metric spaces

1. Continuous functions in metric spaces

Let M, N be metric spaces with metrics dm and dn respectively.
Let f be a function from M to N.

Suppose that if (an) is any convergent sequence in M, then (f(an)) is a convergent sequence in N.

Show that f is a continuous function.

Note that the question does not say that (an) --> a implies (f(an)) --> f(a). If it did then I could solve the question. But, as it is currently stated, I do not know the answer.

2. You know that f is continuous if and only if $\displaystyle a_n \rightarrow a \Rightarrow f(a_n) \rightarrow f(a)$, right?

I would assume that $\displaystyle f(a_n) \rightarrow c$ and then show that $\displaystyle c = a$. Or assume $\displaystyle f(a_n)$ approaches a limit distinct from $\displaystyle a$ and then derive a contradiction.

3. what happened to plato post??

4. Hint: Suppose that $\displaystyle a_n\to a$ and that $\displaystyle f(a_n)\to b$. What can you say about the limit of the sequence $\displaystyle a_1,a,a_2,a,a_3,a,\ldots$?

5. Originally Posted by southprkfan1
Let M, N be metric spaces with metrics dm and dn respectively.
Let f be a function from M to N.

Suppose that if (an) is any convergent sequence in M, then (f(an)) is a convergent sequence in N.

Show that f is a continuous function.

Note that the question does not say that (an) --> a implies (f(an)) --> f(a). If it did then I could solve the question. But, as it is currently stated, I do not know the answer.
Alternatively to Opalg's (this is true in any topological space) but let $\displaystyle N$ be an arbitrary neighborhood of $\displaystyle f(a)$ then $\displaystyle f^{-1}(N)$ is a neighborhood of $\displaystyle a$ (by $\displaystyle f$'s continuity) and so we see that $\displaystyle f^{-1}(N)$ is an open set containing $\displaystyle a$ and since $\displaystyle a_n\to a$ it contains all but finitely many of $\displaystyle \{a_n\}_{n\in\mathbb{N}}$ so then $\displaystyle ff^{-1}(N)\subseteq N$ contains all but finitely many values of $\displaystyle f(a_n)$.

6. Originally Posted by Drexel28
$\displaystyle f^{-1}(N)$ is a neighborhood of $\displaystyle a$ (by $\displaystyle f$'s continuity) .
But the point is to show that $\displaystyle f$ is continuous.
So we cannot assume that it is, can you?

7. Originally Posted by Plato
But the point is to show that $\displaystyle f$ is continuous.
So we cannot assume that it is, can you?
Oops! Misread the question. And well...I probably shouldn't give the full solution...but I do want to mention that this is true whenever the domain is a first countable space.

8. Originally Posted by Opalg
Hint: Suppose that $\displaystyle a_n\to a$ and that $\displaystyle f(a_n)\to b$. What can you say about the limit of the sequence $\displaystyle a_1,a,a_2,a,a_3,a,\ldots$?
Thanks for hint, I see how it implies that (f(an)) converges to f(a).

Just out of curiosity, is this really the best way to go about it? I would think there's a more "direct" route to take. Maybe I'm wrong.

Either way, thanks a lot!

9. Originally Posted by southprkfan1
Thanks for hint, I see how it implies that (f(an)) converges to f(a).

Just out of curiosity, is this really the best way to go about it? I would think there's a more "direct" route to take. Maybe I'm wrong.

Either way, thanks a lot!
There is, a way dealing with open balls around $\displaystyle f(x)$ and $\displaystyle x$. Try writign out what continuity means and see if something doesn't "strike" you

10. Originally Posted by southprkfan1
I see how it implies that (f(an)) converges to f(a). Just out of curiosity, is this really the best way to go about it? I would think there's a more "direct" route to take.
That is the very defintion of continuity.
How can that be inproved upon?

11. Originally Posted by Drexel28
There is, a way dealing with open balls around $\displaystyle f(x)$ and $\displaystyle x$. Try writign out what continuity means and see if something doesn't "strike" you
Well, I'll tell you off the bat I have know idea what "strike" represents

But, here's what I was working on before, and you can feel free to give me your thoughts.

Suppose f is not continuous:

Then, there exists e>0 and a xn (n=1,2,3,...) in M where

dM(xn, a) < 1/n but dN(f(xn), f(a)) > e

clearly (xn) converges to a, so by assumption (f(xn)) is a convergent sequence. Say it converges to f(x)

Thus, for n large enough, we have dN(f(xn), f(x))< e + d(f(x),f(a)) and dM(an, a)<1/n

but d<(xn, a) < 1/n --> e < dN(f(xn), f(a)) <= d(f(xn), f(x)) + d(f(x),f(a))

--> dN(f(xn), f(a)) > e - dN(f(x),f(a))

--> e + d(f(x),f(a)) > e + dN(f(x),f(a))

--> d(f(x),f(a)) > 0

--> f(x) /= f(a)

And I'm stuck
(well, not really, I could use the last posters hint to say f(x) = f(a), but in that case all this crap above would be unnecessary)

12. Originally Posted by southprkfan1
Well, I'll tell you off the bat I have know idea what "strike" represents

But, here's what I was working on before, and you can feel free to give me your thoughts.

Suppose f is not continuous:

Then, there exists e>0 and a xn (n=1,2,3,...) in M where

dM(xn, a) < 1/n but dN(f(xn), f(a)) > e

clearly (xn) converges to a, so by assumption (f(xn)) is a convergent sequence. Say it converges to f(x)

Thus, for n large enough, we have dN(f(xn), f(x))< e + d(f(x),f(a)) and dM(an, a)<1/n

but d<(xn, a) < 1/n --> e < dN(f(xn), f(a)) <= d(f(xn), f(x)) + d(f(x),f(a))

--> dN(f(xn), f(a)) > e - dN(f(x),f(a))

--> e + d(f(x),f(a)) > e + dN(f(x),f(a))

--> d(f(x),f(a)) > 0

--> f(x) /= f(a)

And I'm stuck
(well, not really, I could use the last posters hint to say f(x) = f(a), but in that case all this crap above would be unnecessary)
How about this. I am going to assume a lot of stuff and let you generalize! Mainly that you call a function $\displaystyle f$ continuous at $\displaystyle x$ if $\displaystyle \lim_{\xi\to x}f(\xi)=f(x)$.

So, now suppose that $\displaystyle x_n\to x\implies f(x_n)\to f(x)$ but $\displaystyle \lim_{\xi\to x}f(\xi)=y\ne f(x)$. Then, since any metric space is Hausdorff we may find open sets $\displaystyle O_y,O_{f(x)}$ such that $\displaystyle y\in O_y,f(x)\in O_{f(x)}$ but $\displaystyle O_y\cap O_{f(x)}=\varnothing$, and since $\displaystyle f(x_n)\to f(x)$ we have that $\displaystyle O_{f(x)}$ contains all but finitely many points of $\displaystyle f(x_n)$. Now the definition for $\displaystyle \lim_{\xi\to x}f(x)=y$ means that every neighborhood $\displaystyle E$ of $\displaystyle y$ there exists a neighborhood $\displaystyle N$ of $\displaystyle x$ such that $\displaystyle f\left(N-\{x\}\right)\subseteq E$. But, given any neighborhood $\displaystyle N$ of $\displaystyle x$ we have that there exists all but finitely many points of $\displaystyle \{x_n\}$ and so it follows that $\displaystyle f\left(E-\{x\}\right)\cap O_{f(x)}\ne\varnothing\implies f\left(E-\{x\}\right)\nsubseteq O_y$. This is clearly contradicts that $\displaystyle \lim_{\xi\to x}f(\xi)=y$. It follows that $\displaystyle \lim_{\xi\to x}f(\xi)=f(x)$.

Note: It is really late here and I may have a) made a mistake b) missed a much easier method. Also, I assumed that the limit exists...and it is easy to prove that it must.