Cardnality of Infinite Sets

[kingwinner]

1) Find the cardnality of the set of continuous functions from R to R.

Let's consider a simplified version: for the cardnality of the set of functions from R to R, I can compare it with the set of characteristic functions of subsets of R and conclude that they both have cardnality 2^c

But when the word "continuous" is inserted, how can I find its cardnality?

Thanks for any help!

 

[Dick]

If f is a continuous function and you know the value of f(q) for all rational q, do you know the value of f for any real value?

 

[kingwinner]

If f is a continuous function and you know the value of f(q) for all rational q, do you know the value of f for any real value?

I think it would be hard to tell specifically. Since f is continuous, the values can't go too far, but there is still a wide possible range. Another thing is that we may not know the value of f(q) for q rational

 

[kingwinner]

Also,
since |{functions from R to R}|=2^c,
I think that |{continuous functions from R to R}|<2^c.
This narrows down the possible answers, but I still don't know how to get the cardnality exactly.

 

[Dick]

I think it would be hard to tell specifically. Since f is continuous, the values can't go too far, but there is still a wide possible range. Another thing is that we may not know the value of f(q) for q rational

That was a rhetorical question. Let me put it this way, if you know the value of f for all rationals, then you know the value of f for all reals. Why do I believe this? That means while a general functions have cardinality c^c the cardinality of continuous functions may be less. Is it?

 

[kingwinner]

That was a rhetorical question. Let me put it this way, if you know the value of f for all rationals, then you know the value of f for all reals. Why do I believe this? That means while a general functions have cardinality c^c the cardinality of continuous functions may be less. Is it?

Sorry, I am lost for two reasons...

The question asks for the cardnality of the set of continuous functions from R to R, so shouldn't we not be assuming any further detail. It didn't say that the value of f for all rationals are known. But perhaps I am misunderstanding the question...

Also, why is the answer c^c? I haven't encountered the cardinal number c^c so I don't know what it means. Shoudn't the answer be less than or equal to 2^c?

 

[Dick]

c^c=2^c. Wasn't that what you showed in the first part? What I'm saying is that continuous functions are DETERMINED by their values on the rationals Q. Q is countable.

 

[HallsofIvy]

Sorry, I am lost for two reasons...

The question asks for the cardnality of the set of continuous functions from R to R, so shouldn't we not be assuming any further detail. It didn't say that the value of f for all rationals are known. But perhaps I am misunderstanding the question...

That's not the point. Let x be any real number. Let {x_n} be a sequence converging to x. Since f is continuous at x, $\lim_{n\rightarrow \infty} f(x_n)= f(x)$. That is, if you two continuous functions are equal on all rational numbers, then they are equal on all real numbers. Therefore the cardinality of the set of all continuous functions from R to R cannot be larger than the cardinality of the set of all functions from Q to R.

Also, why is the answer c^c? I haven't encountered the cardinal number c^c so I don't know what it means. Shoudn't the answer be less than or equal to 2^c?

Where did you see that the answer was c^c?