I am stuck on this question from a problem set. The question itself has two parts so I post them under the same topic (they're pretty much related to each other too - at least in the book). I hope it's ok.
Q1: Let $f:[a,\infty] \rightarrow \mathbb R$ derivative function. \lim_{x\rightarrow\infty} f'(x) = \infty. Show that $f$ is not uniformly continuous on $[a,\infty]$
Attempts:
- I tried to show that if f'(x) isn't blocked, the function cannot be uniformly continuous, but it only works in the opposite direction.
- Showing that $\lim_{x\rightarrow\infty} f(x)$ doesn't exist doesn't help me either.
Q2: Let $f:\mathbb R \rightarrow \mathbb R$ continuous function. $\lim_{x\rightarrow\infty} f(x) = \lim_{x\rightarrow-\infty} f(x) = L$. Show that $f$ has either minimum or maximum in $\mathbb R$
Attempts:
- Exists $M_1$ (for $\lim_{x\rightarrow\infty} f(x)$) and $M_2$ (for $\lim_{x\rightarrow-\infty} f(x)$) and if function gets max or min in $[M_1, M_2]$ which is equal to L, then we're done.
Thanks in advance!