Suppose $f$ is a real-valued function and $f(x) \geq 1$ for every real $x \in [0,1]$.
Why we can always find a unique positive integer $n$ such that $2^{n} \leq f(x) < 2^{n+1}$ ?
Suppose $f$ is a real-valued function and $f(x) \geq 1$ for every real $x \in [0,1]$.
Why we can always find a unique positive integer $n$ such that $2^{n} \leq f(x) < 2^{n+1}$ ?
The intervals $[2^n,2^{n+1})$ cover the entire real half (existence) line $y\geq 1$ and no two of them intersect (uniqueness). So any point on the graph of the function must intersect one of these since $f(x)\geq 1$. You don't need any hypothesis on the domain of the function.
You can't do it for $f(x)=3x+1$.