# x-2-x-x-x-x-x-times-taking-derivate-2x-1-1-1-1-x-times-2x-x-x-0-2-1-where-the-problem-

Question Number 2851 by 123456 last updated on 28/Nov/15
$${x}^{\mathrm{2}} ={x}+{x}+{x}+\centerdot\centerdot\centerdot+{x}\:\left({x}\:\mathrm{times}\right) \\$$$$\mathrm{taking}\:\mathrm{derivate} \\$$$$\mathrm{2}{x}=\mathrm{1}+\mathrm{1}+\mathrm{1}+\centerdot\centerdot\centerdot+\mathrm{1}\:\left({x}\:\mathrm{times}\right) \\$$$$\mathrm{2}{x}={x}\:\left({x}\neq\mathrm{0}\right) \\$$$$\mathrm{2}=\mathrm{1} \\$$$$\mathrm{where}\:\mathrm{the}\:\mathrm{problem}? \\$$
Commented by Yozzi last updated on 29/Nov/15
$${That}\:{first}\:{line}\:{assumes}\:{that}\:{x}\:{is}\: \\$$$${a}\:{whole}\:{number}.\:{If}\:{x}\:{is}\:{variable}\:{and}\:{real} \\$$$${and}\:{we}\:{let}\:{x}\:{be}\:{irrational}\:\left({that}\:\right. \\$$$$\left.{is}\:{not}\:{of}\:{the}\:{form}\:\sqrt{{r}}\:,\:{r}\in\mathbb{Q}\right) \\$$$${x}^{\mathrm{2}} \:{cannot}\:{be}\:{expressed}\:{as}\:{a}\:{rational} \\$$$${number}\:{of}\:{the}\:{form}\:\frac{{a}}{{b}}\:{since} \\$$$${x}^{\mathrm{2}} =\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{b}}+…+\frac{\mathrm{1}}{{b}}\:\:\left({a}\:{times}\right) \\$$$${is}\:{a}\:{contradiction}.\: \\$$$$\\$$$${Another}\:{non}−{example}\:{could}\:{be}\: \\$$$${that}\:{if}\:{x}=\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow{x}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}}\:{but}\:{what}\:{is} \\$$$$\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}+…+\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\left(\frac{\mathrm{1}}{\mathrm{2}}\:{times}\right)? \\$$$$\\$$$${x}\:{must}\:{be}\:{a}\:{whole}\:{number}.\:{So}\:{that} \\$$$${y}={x}^{\mathrm{2}} \:{is}\:{not}\:{continuous}\:{on}\:{any}\: \\$$$${sub}−{interval}\:{of}\:{the}\:{real}\:{axis}, \\$$$${except}\:{at}\:{points}. \\$$$${The}\:{derivative}\:{of}\:{the}\:{y}={x}^{\mathrm{2}} \:{does} \\$$$${not}\:{exist}\:{for}\:{x}\notin\mathbb{Z}\:{since}\:{y}={x}^{\mathrm{2}} \:{is}\: \\$$$${undefined}\:{at}\:{such}\:{x}. \\$$
Commented by Yozzi last updated on 29/Nov/15
$${Let}\:{f}\left({x}\right)=\begin{cases}{{x}^{\mathrm{2}} \:\:\:{x}\in\mathbb{Z}^{+} }\\{\mathrm{0}\:\:\:\:{otherwise}}\end{cases} \\$$
Commented by Rasheed Soomro last updated on 29/Nov/15
$$\mathcal{V}{ery}\:\mathcal{N}{ice}! \\$$