Jtdylktuy

(newbie here)

I'm trying to prove that any open real interval is uncountable.

I try to proove this with a bijective function (a,b) --> R. Is there a further criteria for this than having a bijective function?
If not, would it be possible to simply use an easy function like y=x, where x is an element in the real open interval (a,b) and y is an element of R.

Note that what Cantor's diagonal argument shows directly is not that $mathbb R$ itself is uncountable, but that the particular open interval $(0,1)$ is uncountable.

Since $(0,1)subseteq mathbb R$, this immediately implies that $mathbb R$ is also uncountable, but for your purpose it is easier to aim directly for $(0,1)$.
Namely, if you have a different open interval $(a,b)$, you can map it bijectively to $(0,1)$ by a simple linear interpolation:

$$ x in (a,b) mapsto frac{x-a}{b-a} in (0,1) $$

would it be possible to simply use an easy function like y=x, where x is an element in the real open interval (a,b) and y is an element of R.

This does not really specify a function. In order to have a function you need to have a particular $y$ for each $xin(a,b)$, and a rule for which $y$s match up to which $x$s.

If you have only proved that $Bbb R$ is uncountable and want to prove that any interval $(a,b)$ is uncountable you can use any bijection between $(a,b)$ and $Bbb R$. $y=x$ does not supply that bijection because all the elements outside $(a,b)$ do not have an image. The easiest way is to find a function that stretches the interval and any such function will do.

Often these are done by a chain of bijections. You can biject any open interval with any other open interval using a linear function, so instead of the general $(a,b)$ you can use an interval that is convenient, like $(0,1)$. One common one is to use the interval $(-frac pi 2, frac pi 2)$ and the function $y=tan x$