$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{mathematical}\:....{analysis}..... \\$$$$\:\:{suppose}\:\:\:\:{f}\::\left[{a}\:,\:{b}\right]\rightarrow\mathbb{R}\:{is}\:{a}\:{function} \\$$$$\:\:\:{and}\:\:\:\alpha:\left[{a}\:,\:{b}\right]\overset{\alpha\nearrow} {\rightarrow}\mathbb{R}\:\left(\alpha\:{is}\:{an}\:{increasing}\:{function}\right. \\$$$$\left.\:{on}\:\left[{a}\:,\:{b}\right]\right)\:\:{meanwhile}\:\alpha\:{is}\:{continuous}\:{at}\:{y}_{\mathrm{0}} \: \\$$$$\:\:{where}\:\:\:{a}\leqslant{y}_{\mathrm{0}} \leqslant{b}\:\:.\:{defining}\: \\$$$$\:\:\:{f}\left({x}\right)=\begin{cases}{\:\mathrm{1}\:\:\:\:\:\:\:\:\:{x}={y}_{\mathrm{0}} }\\{\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:{x}\neq{y}_{\mathrm{0}} }\end{cases} \\$$$$\:\:\:\:{prove}\:\:{that}\::\:{f}\in\:\mathscr{R}\:\left(\alpha\right)\:.... \\$$$$\:\:\:\:{Hint}:\:{f}\in\mathscr{R}\:\left(\alpha\right)\:\Leftrightarrow\:\forall\:\epsilon>\mathrm{0}\:\exists\:{P}_{\epsilon} \:;\:{U}\left({P}_{\epsilon} ,{f},\alpha\right)−{L}\left({P}_{\epsilon} ,{f},\alpha\right)<\epsilon \\$$$$\:\:\:\:{Reimann}\:\:{criterion}\:.... \\$$