I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$?
Is there such function, or am I wasting my time? Thanks!
I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$?
Is there such function, or am I wasting my time? Thanks!
Pick an $a\gt 0$. Then by the Mean Value Theorem, there exists a point $r$, $a\lt r\lt a+M$, such that $f'(r)=\frac{1}{M}(f(a+M)-f(a))$. That means that $f(a+M) - f(a) = Mf'(r) \geq Mc$ hence for every $M\gt 0$ we have $f(a+M) \geq f(a)+Mc.$
What happens as $M\to\infty$?
If $f'(x)>c$ for every $x$, then $f(x)-f(0) > c x$ for every $x>0$. Hence $\liminf_{x \to +\infty} f(x) \geq \liminf_{x \to +\infty} \left(f(0)+cx \right) = +\infty.$