Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Let $x \in \mathbb{R}$ and $a$ (either positive or negative or zero) be a constant also in the domain of $f$. I would like to be able to bound the product $f(x + a)f(x)$.
Is it true that for any continuous function and any points $x$ and $a$ at least one of the following statements will be true:
- $f(x)^2 \leq |f(x + a)f(x)| \leq f(x + a)^2$
- $f(x+a)^2 \leq |f(x + a)f(x)| \leq f(x)^2$
- $f(x+a)^2 \leq |f(x + a)f(x)| \leq f(x+a)^2$
- $f(x)^2 \leq |f(x + a)f(x)| \leq f(x)^2$
Graphically, it makes sense for me but I am struggling to prove it mathematically.