Mathematical-Analysis-f-R-R-is-diffrentiable-function-f-and-f-has-no-common-zero-on-R-prove-that-the-following-set-is-finite-X

Question Number 211486 by mnjuly1970 last updated on 10/Sep/24

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:{Mathematical}\:\:\:\:\:{Analysis}… \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:{f}:\mathbb{R}\:\rightarrow\:\mathbb{R}\:{is}\:{diffrentiable}\:{function}, \\ $$$$\:\:\:\:\:{f}\:\:{and}\:\:{f}\:'\:,\:{has}\:{no}\:{common}\:{zero} \\ $$$$\:\:\:\:\:{on}\:\:\mathbb{R}\:. \\ $$$$\:\:\:\:\:\:{prove}\:{that}\:{the}\:{following}\:{set}\:{is}\:{finite}. \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{X}\:=\:\left\{\:{x}\:\in\:\left[\mathrm{0}\:,\:\mathrm{1}\:\right]\:\mid\:{f}\left({x}\right)=\mathrm{0}\:\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−−−−− \\ $$

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