Thread: Real valued function on [0,1]

1. Real valued function on [0,1]

I'm stuck on the following question:

Let A be the set of all real-valued functions on [0,1]. show that there does nor exist a function from [0,1] onto A.

I don't even know where to start.

2. Originally Posted by lllll
Let A be the set of all real-valued functions on [0,1]. show that there does nor exist a function from [0,1] onto A.
This is a rather advanced theorem.
So I have no way of knowing what you have already proved.
But here are the lemmas you will need.

1) There is an injection $f:X \to Y$ if and only if there is a surjection $g:Y \to X$.
2) There is no surjection $f:X \to P(X)$, (power set of $X$).
3) If $2^X = \left\{ {f:X \mapsto \{ 0,1\} } \right\}$, the set of functions from $X$ having only two values: 0 or 1.
Then $2^X \sim P(X)$
Now clearly the set $2^{[0,1]} \subseteq A = \left\{ {f:[0,1] \to \mathbb{R}} \right\}$

If there were $\varphi :\left[ {0,1} \right]\xrightarrow{{surj}}A$ then
$\varphi :\left[ {0,1} \right]\xrightarrow{{surj}}2^{[0,1]} \sim P\left( {\left[ {0,1} \right]} \right)$.