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Probably wrong heuristic and assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$.

Assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$.

Probably wrong heuristic and assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$. Or approximating $f^{2}(1/t)/t^{2}$ by continuous functions.

Probably wrong heuristic and assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$.

Probably wrong heuristic and assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$.

Probably wrong heuristic and assuming lots of stuff, just to get the ball rolling:

$$\frac{1}{f(x)}\int_{x}^{\infty}f^{2}(t)dt=\frac{1}{xf(x)}\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

So we claim that

$$(xf(x))^{2}\sim \frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}.$$

If $f^{2}(1/t)/t^{2}$ was continuous, then

$$\frac{\int_{0}^{\frac{1}{x}}f^{2}(\frac{1}{t})\frac{1}{t^{2}}dt}{\frac{1}{x}}\sim f^{2}(x)x^{2}\frac{1/x}{1/x}=f^{2}(x)x^{2}.$$

So maybe getting a counterexample by making $f^{2}(1/t)/t^{2}$ not well-behaved eg. blowing up at $t=0$ i.e. $f(x)>\frac{1}{x}$. Or approximating $f^{2}(1/t)/t^{2}$ by continuous functions.