let $\displaystyle f(x)$be continuous on $\displaystyle [ 0,\infty )$. Define $\displaystyle L=\{z\in R \ni \exists$ $\displaystyle a$ $\displaystyle sequence$ $\displaystyle x_n \rightarrow \infty$ $\displaystyle and$ $\displaystyle f(x_n) \rightarrow z$ $\displaystyle as$ $\displaystyle n \rightarrow \infty\}$
Prove a) $\displaystyle L$ is closed b)$\displaystyle L$ is connected c)$\displaystyle L$ is an interval
2. If you recall the definitions of limes superior and limes inferior, prove $\displaystyle L=[{\rm liminf}_{\infty}f,{\rm limsup}_{\infty}f]$.