Does there exist any continuous probability density function $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is increasing on $[1,+\infty)$?

Question

Does there exist any continuous probability density function $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is increasing on $[1,+\infty)$?

In my research I dealt to the above question and I think the answer should be negative because of the following reason:

Since $f$ is continuous then we have $f=F^\prime$ ($F$ is its CDF), Moreover if $f=F^\prime$ is increasing along with $\lim_{x\to\infty}F(x)=1$ then $F$ must be decresing and it is a contradiction with this fact that "$F$ is always non-decreasing". I am very pleased if anyone can help me.

Answer 1

If $f(a) >0$ for some $a$ then we get $1=\int f(x)\, dx \geq \int_a^{\infty} f(a) dy=\infty$, a contradiction. Thus $f(a)=0$ for all $a$ but then $\int f(y)dy\neq 1$.