Let $f:\mathbb{R}\rightarrow \left ( 0,\infty \right )$ be a twice differentiable function such that $f\left ( 0 \right )=1$ and $\int_{a}^{b}f\left ( x \right )dx=\int_{a}^{b}{f}'\left ( x \right )dx$, for all $a,b \in \mathbb{R}$, with $a\leq b$. Which of the following statements is false?
- $f$ is one to one
- The image of $f$ is compact
- $f$ is unbounded
- There is only one such function.