In the previous real analysis sessions in my university , a question came as follow: A function f defined for x>0 is increasing such that g(x) =f(x) /x is decreasing. Prove that f is continous.
I tried to work using the ε-δ definition of continuity and to make use of the variations of f by trying to find the limit just before and after any strictly positive number a, but I am not finding a way to bound the difference between f(x) and f(a). I hope any could give me a hint or a way to start with.
Here's a hint:
Suppose that $0<x<a$. Since $g$ is decreasing, we have
$$\frac{f(a)}{a}\leq\frac{f(x)}{x}$$
However, $f$ is increasing, and since $x<a$, $f(x)\leq f(a)$, so
$$\frac{f(a)}{a}\leq\frac{f(x)}{x}\leq\frac{f(a)}{x}$$
What happens as $x\to a^-$?
A similar argument can be done when $x\to a^+$. This justifies that $g$ is continuous. Then, just justify that $f$ is continuous.