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2026-04-07 23:13:45.1775603625
Continuity and Intermediate Value Theorem.
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If $ f(1) \neq 0$ and $f$ is supposed to be continuous at $1$ then
$$\lim\limits_{x \to 1} f(x) f\left(\frac{1}{x}\right) = \left(\lim\limits_{x \to 1} f(x)\right)\left(\lim\limits_{x \to 1} f\left(\frac{1}{x}\right)\right) =f^2(1) \gt 0.$$
In contradiction with the given hypothesis $f(x) f\left(\frac{1}{x}\right) \lt 0$ which implies $ \lim\limits_{x \to 1}f(x) f\left(\frac{1}{x}\right) \le 0$. Therefore $f(1) =0$.