Is there infinite number between two integers?

Question

With more specific saying,

if there is a function that is in continuum from a to b, a and b are real numbers, and bijective to its range, is the number of the function's range infinite?

If it is, are all continuums infinite?

Answer 1

You have to be careful... is your function continuous at $x=a$ and $x=b$?

If this function only has to be continuous on $(a,b)$ then, yes, you can find continuous 'infinite functions'.

For example,

$$f(x)=\tan\left(\frac{\pi}{b-a}\left(x-\frac{a+b}{2}\right)\right)$$

is a bijection from $(a,b)$ to $\mathbb{R}$.

If you are continuous on the closed interval $[a,b]$ then the answer is no because the image of a compact set under a continuous function is compact. In other words the best you can do is map $[a,b]$ to another interval $[a',b']\subset\mathbb{R}$.

You can construct a sequence, $\{f_N\}_{N\geq1}$, of continuous and bijective functions from $[a,b]$ to $[-N,N]$ but this sequence does not converge to a continuous function.