Question Number 224568 by fkwow344 last updated on 19/Sep/25
$$\sqrt{{x}}={a\begin{cases}{{a}\in\mathbb{Z}=\mathrm{0}}\\{{a}\notin\mathbb{Z}=\mathrm{1}}\end{cases}} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{3}} \:\sqrt{{x}}{dx}=? \\ $$
Commented by Ghisom_ last updated on 19/Sep/25
$$\mathrm{what}\:\mathrm{does}\:\mathrm{this}\:\mathrm{mean}? \\ $$$$\mathrm{maybe}\:\begin{cases}{{f}\left({x}\right)=\mathrm{0};\:\sqrt{{x}}\in\mathbb{Z}}\\{{f}\left({x}\right)=\mathrm{1};\:\sqrt{{x}}\notin\mathbb{Z}}\end{cases} \\ $$$$\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}=\mathrm{3} \\ $$$$\mathrm{because} \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:{f}\left({x}\right)\:=\mathrm{1}\:\wedge\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}\:{f}\left({x}\right)\:=\mathrm{1}} \\ $$