Is Continuity Defined by the Behavior of Function Over Closure of Sets?

[symplectic_manifold]

Hi!
Please, give me some guidance in solving this problem.

Let [itex]f:{\mathbb{R}}^n\longrightarrow{\mathbb{R}}^m[/itex].
Show that f is continuous iff for all [itex]M\subset{\mathbb{R}}^n[/itex] the inclusion [itex]f(closM)\subseteq{clos{}f(M)}[/itex] holds.(closM denotes the closure of the set M)

Please, ask me some guiding questions to lead me in the right direction straightaway.

 

[matt grime]

First off you should tell us what your definition of continuous is so that we do not use one of the many equivalent conditions that we aren't supposed to (eg inverse image of open set is open).

 

[symplectic_manifold]

The definition we mainly use is:

The mapping [itex]f:X\longrightarrow{Y}[/itex] is continuous in [itex]x_0\in{X}[/itex] iff for all [itex]\epsilon>0[/itex] there exists [itex]\delta>0[/itex] such that for all [itex]x\in{X}[/itex] with[itex]d_X(x,x_0)<\delta[/itex] the following inequality holds: [itex]d_Y(f(x),f(x_0))<\epsilon[/itex]

I forgot to tell about the hint. It says we should use the closure of the set of the terms of some partial sequence, which should then be mapped by f. I can't get anything out of it.
I found an alternative proof of the above proposition (in Djedonne's "Analysis"), but it uses adherent points of M...which I don't think we are supposed to use, because we didn't explicitly treated these concepts.

 

[matt grime]

So, you're specifically talking about this as a metric topology.

Do you know the definition that a function is continuous at x if and only if for all sequences x_n tending to x then f(x_n) tends to x. You shuold try to prove this is equivalent to your commonly used definition of continuity.

 

[symplectic_manifold]

OK, but will it bring closure with it into the game?

 

[matt grime]

Again, how easily useful this hint is will depend upon how *you* define the closure. There are at least 4 different (but equivalent) ways of doing this.

 

[symplectic_manifold]

Our definition of closure is:
The closure of a set is the union of this set with all its limit points.

How can I formally connect the definition of continuity with the above def. of closure to get a statement about some terms of a subsequence in X converging to some x_0? What should happen when the closure of the set of these terms (hint) is mapped under f? Should there be a contradiction?

 

[matt grime]

Let me retell you what you already know.

Let x_n be a series in M that tend to x, ie a point in the closure.

What happnens when we apply f?

Now, every point in the closure of M is the limit of such a sequence.

Hence the result is true.