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Question Number 165339 by mnjuly1970 last updated on 30/Jan/22

$$$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$\mathrm{L}et,f:\mathbb{R}\to \mathbb{Q}isacontinuous$$$$function.provethat{}^{″}f{}^{″}isa$$$$constantfunction.◼m.n$$$$\ast Adoptedfrommathematicalanalysisbook\ast$$$$--------------$$$$$$

Answered by mahdipoor last updated on 30/Jan/22

$$proofbycostradiction:$$$$iffor\mathrm{\forall }a<b,f\left(a\right)\ne f\left(b\right)$$$$\mathrm{\exists }k\in {\mathrm{Q}}^{{}^{\prime }}thatbetweenf\left(a\right)andf\left(b\right)$$$$\Rightarrow fiscontinuousfunction,$$$$mediumvaluetheorem:$$$$\mathrm{\exists }c\in [a,b]thatf\left(c\right)=k\in {\mathrm{Q}}^{{}^{\prime }}$$$$butf:\mathrm{R}\to \mathrm{Q}(andcannotf\left(c\right)\in {\mathrm{Q}}^{{}^{\prime }})$$$$therefor,for\mathrm{\forall }a<b,f\left(a\right)=f\left(b\right);$$$$thismeansthatfisconstantfunction.$$

Commented by mnjuly1970 last updated on 30/Jan/22

$$bravosir...zendebashid$$

Commented by mahdipoor last updated on 30/Jan/22

$$payandebashid:)$$

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