Does $f(2x) \in Θ(f(x))$ always hold?

Question

If $f(x)$ is continuous and increasing positively, does $f(2x) \in Θ(f(x))$?

I am convinced that this is false but I am stuck on the proof.

$$0 \le c_1 f(x) \le f(2x) \le c_2 f(x)$$ $$0 \le c_1 \le \frac{f(2x)}{f(x)} \le c_2$$

The inequality on the right-side seems interesting (for disproving this). But I cannot think of a $f(x)$ function where this inequality wouldn't hold!

Thanks in advance for any ideas or pointers.

Answer 1

To disprove it, all you need is a counterexample. Try something exponential like $f(x) = 2^x$.